Research ArticleFinitely Generated Modules over Group Rings ofa Direct Product of Two Cyclic Groups

Ahmed Najim and Mohammed Elhassani Charkani

Department of Mathematics and Informatics Faculty of Science Dhar Mahraz Sidi Mohamed Ben Abdellah University30000 Fez Morocco

Correspondence should be addressed to Ahmed Najim najimsefrougmailcom

Received 27 August 2014 Accepted 15 November 2014 Published 1 December 2014

Academic Editor Zhongshan Li

Copyright copy 2014 A Najim and M E CharkaniThis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Let119870 be a commutative field of characteristic 119901 gt 0 and let119866 = 1198661times1198662 where119866

1and119866

2are two finite cyclic groupsWe give some

structure results of finitely generated 119870[119866]-modules in the case where the order of 119866 is divisible by 119901 Extensions of modules arealso investigated Based on these extensions and in the same previous case we show that119870[119866]-modules satisfying some conditionshave a fairly simple form

1 Introduction

Let 119870 be a field of characteristic 119901 gt 0 and let 119866 be a finitegroupThe study of119870[119866]-modules in the casewhere the orderof 119866 is divisible by 119901 is a very difficult task When 119866 is afinite abelian 119901-group we find in [1] the following statementa complete classification of finitely generated 119870[119866]-modulesis available only when119866 is cyclic or equal to119862

2times1198622 where119862

2

is the cyclic group of order 2 In [2] we find this classificationin these two cases Still more in the case where the Sylow 119901-subgroup 119875 of 119866 is not cyclic the groups 119866 such that 119901 = 2

and 119875 is dihedral semidihedral or generalized quaternionare the only groups for which we can (in principle) classifythe indecomposable 119870[119866]-modules (see [2]) These reasonsjust cited show the importance of the study of119870[119866]-moduleswhen119866 is of order divisible by 119901 and equal to a direct productof two cyclic groups

Now let 119870 be a commutative field of characteristic 119901 gt 0and let 119866 = 119866

1times 1198662 where 119866

1and 119866

2are two finite cyclic

groups Let 119872 be a finitely generated 119870[119866]-module When119872 is considered as a module over a subalgebra119870[119867] of119870[119866]for a subgroup119867 of the group 119866 we write119872darr

119867

In Section 2 we show that if 1198661is a cyclic 119901-group and

the characteristic of 119870 does not divide the order of 1198662 then

we can have a complete system of indecomposable pairwisenonisomorphic 119870[119866]-modules In the rest we assume that

1198661= ⟨120590

1⟩ and 119866

2= ⟨120590

2⟩ are cyclic 119901-groups Under

conditions that 119872darr1198661

is a free 119870[1198661]-module and that

119872(1205901minus 1)119872 is a free 119870[119866

2]-module we show that 119872 is

a free 119870[119866]-module We also show that if 1205902is of order 119901119899

119899 = 0 and 1198672is the subgroup of 119866

2generated by 120590119901

119899minus119903

2

with 0 lt 119903 le 119899 then under certain conditions 119872 is a free119870[1198661times1198672]-moduleThe fact that119872darr

1198661

must be a free119870[1198661]-

module is one of these conditions and exactly in the end ofthis section we give a result that shows when this conditionis satisfied In Section 3 and always in the case where 119866

1and

1198662are cyclic 119901-groups we show that under some conditions

119870[119866]-modules have a fairly simple form But in case 119901 = 21198661and119866

2are two cyclic groups of respective orders 2 and 2119899

119899 = 0 these modules have this simple formwithout any otherassumptions other than that they must be finitely generatedover 119870[119866]

2 Free 119870[119862119901119898 times 119862

119901119899]-Modules of Finite Rank

Throughout this paper rings are assumed to be commutativewith unity We begin this section by giving a weak version ofNakayamarsquos lemma with an elementary proof

Lemma 1 (Nakayama) Let 119866 be a 119901-group with p odd 119877a ring of characteristic 119901119896 where 119896 is a natural number

Hindawi Publishing CorporationAlgebraVolume 2014 Article ID 256020 5 pageshttpdxdoiorg1011552014256020

2 Algebra

119872 an 119877[119866]-module (not necessarily finitely generated) and119873a submodule of119872 and 120590 isin 119866 Then one has the following

(1) if (120590 minus 1)119872 = 119872 then119872 = 0

(2) if119872 = (120590 minus 1)119872 +119873 then119872 = 119873

(3) if 119909119894 119894 isin 119868 are representatives in 119872 of a generating

family of119872(120590 minus 1)119872 then (119909119894)119894isin119868

generate119872

Proof (1) Let 119901119903 be the order of 119866 We have

(120590 minus 1)119901119903

=

119901119903

sum

119894=0

119862119894

119901119903 (minus1)

119894120590119901119903minus119894

= 120590119901119903

+ (minus1)119901119903

+

119901119903minus1

sum

119894=1

119862119894

119901119903 (minus1)

119894120590119901119903minus119894

=

119901119903minus1

sum

119894=1

119862119894

119901119903 (minus1)

119894120590119901119903minus119894

= 119901

119901119903minus1

sum

119894=1

1

119901119862119894

119901119903(minus1 )

119894120590119901119903minus119894

(1)

For 1 le 119894 le 119901119903 minus 1 119901 | 119862119894119901119903 so (1119901)119862119894

119901119903 is a natural number

So

(120590 minus 1)119896119901119903

= ((120590 minus 1)119901119903

)119896

= (119901

119901119903minus1

sum

119894=1

1

119901119862119894

119901119903 (minus1)

119894120590119901119903minus119894)

119896

= 119901119896(

119901119903minus1

sum

119894=1

1

119901119862119894

119901119903 (minus1)

119894120590119901119903minus119894)

119896

= 0

(2)

since 119877 has characteristic 119901119896 Now 119872 = (120590 minus 1)119872 = (120590 minus

1)2119872 = sdot sdot sdot = (120590 minus 1)

119896119901119903

119872 = 0(2) If119872 = (120590minus1)119872+119873 then119872119873 = ((120590minus1)119872+119873)119873 =

(120590 minus 1)(119872119873) and then by (1)119872119873 = 0 and then119872 = 119873(3) If119873 is the submodule generated by 119909

119894 then119872 = (120590 minus

1)119872 +119873 and then by (2) we have119872 = 119873

Remark 2 Lemma 1 remains true if 119901 = 2 and 119877 is ofcharacteristic 2

For a ring 119877 of prime characteristic 119901 and for a cyclicgroup 119866 of order 119901119903 generated by an element 120590 we have thefollowing lemma

Lemma 3 Let 119896 = 1199011199031015840

with 0 lt 1199031015840 le 119903 and let119867 be a subgroupof 119866 generated by 120590119901

119903119896 Then one has 119877[119866](120590 minus 1)119896119877[119866] cong

119877[119867] (as 119877-algebras)

Proof Define

120595 119877 [119866]

(120590 minus 1)119896119877 [119866]

997888rarr 119877 [119867]

120590 997891997888rarr 120590119901119903119896

(3)

where120595 is awell-defined119877-algebra homomorphism It is easyto see that 120595 is surjective As 119877[119866](120590 minus 1)119896119877[119866] and 119877[119867]are finite free modules of the same rank 119896 over 119877 120595 is anisomorphism

Remark 4 With the notation of Lemma 3 119867 is simply thesubgroup of 119866 generated by 120590119901

119903minus1199031015840

Let 119870 be a commutative field of characteristic 119901 gt 0 andlet 119866 be a direct product of two finite groups 119866

1and 119866

2 We

have 119870[119866] = 119870[1198661times 1198662] cong 119877[119866

2] where 119877 = 119870[119866

1]

Assume that 1198661is a cyclic group of order 119901119898 generated

by 1205901and 119901 does not divide the order of 119866

2 119877 = 119870[119866

1] is a

principal Artinian local ring Indeed 119870[1198661] cong 119870[119883](119883 minus

1)119901119898

this isomorphism is induced by the homomorphismΨ 119870[119883] rarr 119870[119866] defined by Ψ(119883) = 120590

1 119870[119883](119883 minus 1)

119901119898

is a principal Artinian local ring with residue field 119870 (up toisomorphism) whosemaximal ideal is generated by119883 minus 1 So119877 is a principal Artinian local ring with residue field119870 (up toisomorphism) whose maximal ideal is generated by 120590

1minus 1

We have119870[119866] cong 119877[1198662] where 119877 is a principal Artinian local

ring of residue field119870The characteristic of119870 does not dividethe order of 119866

2 Under these conditions we can apply [3

Theorem 36] to have a complete system of indecomposablepairwise nonisomorphic119870[119866]-modules

In the remainder of this section we assume that1198661= 119862119901119898

and 1198662= 119862119901119899 are two cyclic groups of respective orders 119901119898

and 119901119899 and are generated respectively by 1205901and 1205902 We have

119870[119866] cong 119877[1198662] As 119877 is a commutative ring and local and

1198662is a 119901-group by [4 Proposition 10 page 239] 119877[119866

2] is a

local ringTherefore119870[119866] is a local ring As119870 is commutativering and local and 119866 is a 119901-group 119870[119866] is a local ring by [4Proposition 10 page 239] So the 119870[119866]-projective modulesare free 119870[119866]-modules

Lemma 5 Let119872 be a119870[119866]-module Then119872(1205901minus 1)119872 is a

119870[1198662]-module (also119872(120590

2minus 1)119872 is a 119870[119866

1]-module)

Proof This lemma is a particular case of a more generalresult (see [5 page 386]) But for this particular case wecan give the following direct proof 119872(120590

1minus 1)119872 is a

(119870[1198661](1205901minus1)119870[119866

1])[1198662]-module andwehave already seen

that (1205901minus 1)119870[119866

1] is the unique maximal ideal of119870[119866

1] and

119870[1198661](1205901minus 1)119870[119866

1] cong 119870 So 119872(120590

1minus 1)119872 is a 119870[119866

2]-

moduleSimilarly we show that119872(120590

2minus 1)119872 is a119870[119866

1]-module

Proposition 6 Let119872 be a free 119870[119866]-module of rank 119897 Then119872(120590

1minus 1)119872 is a free 119870[119866

2]-module and119872(120590

2minus 1)119872 is a

free 119870[1198661]-module of the same rank 119897

Algebra 3

Proof As 119870[119866] is a local ring 119870[119866]-projective modules arefree 119870[119866]-modules and therefore this proposition is onlya particular case of a more general result (see [5 Lemma22]) But for this particular case we can give the followingspecific proof we have 119872 cong (119870[119866])

119897= (119870[119866

1times 1198662])119897 So

119872 cong (119870[1198661][1198662])119897 Then we have

(1205901minus 1)119872 cong (120590

1minus 1) (119870 [119866

1][1198662])119897

cong ((1205901minus 1)119870 [119866

1][1198662])119897

(4)

Hence

119872

(1205901minus 1)119872

cong(119870 [119866

1][1198662])119897

((1205901minus 1)119870 [119866

1][1198662])119897

cong (119870 [1198661][1198662]

(1205901minus 1)119870 [119866

1][1198662])

119897

cong (119870 [1198661]

(1205901minus 1)119870 [119866

1][1198662])

119897

(5)

As 119870[1198661](1205901minus 1)119870[119866

1] cong 119870 (as we have already seen)

119872(1205901minus 1)119872 cong (119870[119866

2])119897 So119872(120590

1minus 1)119872 is a free 119870[119866

2]-

module of rank 119897Similarly we show that 119872(120590

2minus 1)119872 is a free 119870[119866

1]-

module of rank 119897

Proposition 7 Let 119872 be a 119870[119866]-module If 119872darr1198661

is a free119870[1198661]-module and119872(120590

1minus1)119872 is a free119870[119866

2]-module then

119872 is a free 119870[119866]-module

Proof 119877 = 119870[1198661] is a principal Artinian local ring with

residue field 119870 and 1205901minus 1 is a generator of its maximal ideal

119872darr1198661

is a free119877-module and119876 = 119872(1205901minus1)119872 is a projective

119870[1198662]-submodule of 119872(120590

1minus 1)119872 Then 119872 = 119875 oplus 119872

1015840where 119875 is a projective 119877[119866

2]-module and 119875(120590

1minus1)119875 cong 119876 =

119872(1205901minus 1)119872 (according to [3 Proposition 413]) We have

119872

(1205901minus 1)119872

cong119875 oplus119872

1015840

(1205901minus 1) (119875 oplus1198721015840)

cong119875 oplus119872

1015840

(1205901minus 1) 119875 oplus (120590

1minus 1)1198721015840

cong119875

(1205901minus 1) 119875

oplus1198721015840

(1205901minus 1)1198721015840

cong119872

(1205901minus 1)119872

oplus1198721015840

(1205901minus 1)1198721015840

(6)

So1198721015840(1205901minus1)119872

1015840= 0 By Nakayamarsquos lemma and the remark

following it 1198721015840 = 0 Therefore 119872 = 119875 which is projective119877[1198662]-module As 119877[119866

2] cong 119870[119866] is a local ring119872 is a free

119870[119866]-module

Let 119869119894be the Jacobson radical of119870[119866

119894] for 119894 isin 1 2 Note

that if 119870 is of characteristic 119901 (as here) and 1198661015840 is a cyclic 119901-group then the Jacobson radical of 119870[1198661015840] is none other than(120590 minus 1)119870[119866

1015840] where 120590 is a generator of 1198661015840 (see [5 page 122])

Let 119872 be a finitely generated 119870[119866]-module and 119896 anatural number such that 1 le 119896 le 119901119899 As119870[119866] cong 119877[119866

2]119872 is

a 119877[1198662]-module So119872119869

1119872 is a119870[119883](119883 minus 1)119901

119899

-module119872is called of type 119896 if119872119869

1119872 is a free 119870[119883](119883 minus 1)

119896-module(terminology of [6])

Lemma 8 If119872 is a 119870[119866]-module of type 119896 with 119896 = 119901119903 and0 lt 119903 le 119899 and119867

2is the subgroup of119866

2generated by 120590119901

119899minus119903

2 then

1198721198691119872 is a free 119870[119867

2]-module

Proof As 119872 is of type 119896 1198721198691119872 is a free 119870[119883](119883 minus 1)

119896-module Define

120595 119870 [119883]

(119883 minus 1)119896997888rarr 119870[119867

2]

119883 997891997888rarr 120590119901119899minus119903

2

(7)

where120595 is a well-defined119870-algebra homomorphism It is notdifficult to show that120595 is an isomorphism (using an argumentsimilar to that done in the proof of Lemma 3) So119872119869

1119872 is

a free119870[1198672]-module

Theorem 9 Let119872 be a 119870[119866]-module of type 119896 with 1198691198962119872 =

0 and let 1198672be the subgroup of 119866

2generated by 120590119901

119899minus119903

2with

0 lt 119903 le 119899 If 119872darr1198661

is 119877-free and 119896 = 119901119903 then 119872 is a free

119870[1198661times 1198672]-module

Proof 119872 is an 119877[1198662]-module 119877-free We have 119869119896

2119872 = 0 so

(1205902minus 1)119896119872 = 0 and therefore ((120590

2minus 1)119896119877[1198662])119872 = 0 So

119872 is an 119877[1198662](1205902minus 1)119896119877[1198662]-module 119877-free By Lemma 3

119877[1198662](1205902minus 1)119896119877[1198662] cong 119877[119867

2] then 119872 is an 119877[119867

2]-

module 119877-free1198721198691119872 is a free 119870[119883](119883 minus 1)

119896-module soby Lemma 8 this is a free119870[119867

2]-module In conclusion119872 is

a 119870[1198661times 1198672]-module such that

119872darr1198661

is a free 119896 [1198661]-module

119872

1198691119872

is a free 119870[1198672]-module

(8)

So by Proposition 7 119872 is a free119870[1198661times 1198672]-module

In Theorem 9 we assumed that the 119896[119866]-module 119872

satisfies the following condition119872darr1198661

is119877-free So it is usefulto know when this condition is satisfiedThis is the subject ofthe following result

Theorem 10 Let119872 be a 119896[119866]-module and 120590 an element of 119866of order 119901119903 The following conditions are equivalent

(1) 119872darr⟨120590⟩

is free(2) dim

119896(119872) = (119901

119903(119901119903minus 1))dim

119896((120590 minus 1)119872)

(3) dim119896(119872) = dim

119896((120590 minus 1)119872) + dim

119896((120590 minus 1)

119901119903minus1119872)

Proof (1) rArr (2) Assume that 119872darr⟨120590⟩

is free There exists anonzero natural number 119899 such that119872darr

⟨120590⟩cong (119896[⟨120590⟩])

119899 Theendomorphism 120593 of 119872 defined by 120593(119898) = (120590 minus 1)119898 for all

4 Algebra

119898 isin 119872 is nilpotent of nilpotency index 119901119903 and 119896[⟨120590⟩] isan indecomposable 119896[⟨120590⟩]-module Therefore119872 has a basisin which the matrix of 120593 is a Jordan matrix This matrix isformed of 119899 blocks of order 119901119903 all equal to

(

0 1

d dd 1

0

) (9)

So dim119896(119872) = 119901

119903times119899We can easily see that dim

119896((120590minus1)119872) =

119899times(119901119903minus1)Therefore dim

119896(119872) = (119901

119903(119901119903minus1))dim

119896((120590minus1)119872)

(2) rArr (3) Now assume that dim119896(119872) = (119901

119903(119901119903minus

1))dim119896((120590minus1)119872) So dim

119896(119872) = 119901

119903times(dim

119896(119872)minusdim

119896((120590minus

1)119872)) As dim119896(119872)minusdim

119896((120590minus1)119872) is the number of blocks

of the Jordan matrix of 120593 the order of each block is less thanor equal to 119901119903 and dim

119896(119872) is equal to the sum of the orders

of these blocks then the order of each block is 119901119903 Thereforedim119896((120590 minus 1)

119901119903minus1119872) is equal to the number of Jordan blocks

of 120593 So dim119896(119872) minus dim

119896((120590 minus 1)119872) = dim

119896((120590 minus 1)

119901119903minus1119872)

that is dim119896(119872) = dim

119896((120590 minus 1)119872) + dim

119896((120590 minus 1)

119901119903minus1119872)

(3) rArr (1) Assume that dim119896(119872) = dim

119896((120590 minus 1)119872) +

dim119896((120590 minus 1)

119901119903minus1119872) So dim

119896((120590 minus 1)

119901119903minus1119872) is equal to the

number of Jordan blocks of 120593 Therefore the order of eachJordan block is equal to 119901119903 So the modules contained in adecomposition of119872darr

⟨120590⟩as a direct sum of indecomposable

modules are of the form 119896[⟨120590⟩] that is119872darr⟨120590⟩

is free

3 Classification of FinitelyGenerated 119870[119862

119901119898 times 119862

119901119899]-Modules

Use of Module Extensions

Let 119866 be a finite group and let 119877 be a ring Let 1198721and 119872

2

be two 119877[119866]-modules We put 119879 = Hom119877(11987221198721) 119879 has a

natural structure as a (119877[119866] 119877[119866])-bimodule centralized by119877 (see [7 section 25]) Explicitly we have

(120582119891)1198982= 120582 sdot 119891 (119898

2) (119891120582)119898

2= 119891 (120582119898

2)

forall120582 isin 119877 [119866] 1198982isin 1198722 119891 isin 119879

(10)

A derivation 119865 119877[119866] rarr 119879 is an 119877-homomorphismsatisfying

1198651205821015840120582= 1205821015840119865120582+ 1198651205821015840120582 forall120582 120582

1015840isin 119877 [119866] (11)

Derivations from 119877[119866] into 119879 form an 119877-moduleDer(119877[119866] 119879) For 119865 isin Der(119877[119866] 119879) we equip119872

1times1198722with

an 119877[119866]-module structure by

120582 (1198981 1198982) = (120582119898

1+ 119865120582(1198982) 120582119898

2)

forall (1198981 1198982) isin 119872

1times1198722 120582 isin 119877 [119866]

(12)

This 119877[119866]-module is denoted by1198721times1198651198721as in [8]

An extension of 1198722by 119872

1is an 119877[119866]-exact sequence

119874 rarr 1198721rarr 119883 rarr 119872

2rarr 119874 Let 119874 rarr 119872

1

119906

997888rarr

119883V997888rarr 119872

2rarr 119874 and 119874 rarr 119872

1

1199061015840

997888rarr 1198831015840

V1015840997888rarr 119872

2rarr 119874

be a pair of extensions of1198722by1198721These two extensions are

equivalent if there exists an isomorphism of 119877[119866]-modulesΦ 119883 rarr 119883

1015840 such that Φ119900119906 = 1199061015840 and V1015840119900Φ = V

These equivalence classes of extensions form an 119877-moduleExt1119877[119866]

(11987221198721) The 119877[119866]-modules sequence 119874 rarr 119872

1

119894

997888rarr

1198721times1198651198722

119895

997888rarr 1198722rarr 119874 where 119894 and 119895 denote respectively

the canonical injection from1198721to1198721times1198651198722and the second

projection from 1198721times1198651198722to 1198722 is exact The equivalence

class of this sequence is denoted by [1198721times1198651198722]

Remark 11 With the previous notations derivations 119865 andmodules 119872

1times1198651198722play the same role as the cocycles 120572 and

modules1198721times1205721198722defined in [8]

FromProposition 2510 of [7]wehave the following result

Proposition 12 The correspondence 120579 Der(119877[119866] 119879) rarr

1198641199091199051

119877[119866](11987221198721) defined by 120579(119865) = [119872

1times1198651198722] is surjective

whenever1198722is finitely generated and projective as 119877-module

From Theorems 52 and 53 of [9] we have the followingresult

Proposition 13 Let 119866 be a cyclic group of order 119901119903 generatedby an element 120590 119870 a field of characteristic 119901 and 119872 anindecomposable 119870[119866]-module Then119872 is isomorphic to (120590 minus1)119904119870[119866] where 119904 is a natural number strictly less than 119901119903

Lemma 14 Let 119877 be a ring and 119866 = 1198661times 1198662a direct product

of two finite groups Let 119872 be an 119877[119866]-module such that theaction of 119866

1on119872 is trivial and let1198721015840 be an 119877[119866

2]-module If

119872 is isomorphic to1198721015840 as 119877[1198662]-modules and if we extend the

action of 1198662on1198721015840 to 119866 by 120590 sdot 1198981015840 = 1198981015840 forall(1205901198981015840) isin 119866

1times 1198721015840

then119872 is isomorphic to1198721015840 as 119877[119866]-modules

Proof Let 120595 119872 rarr 1198721015840 be an isomorphism of 119877[119866

2]-

modules We extend the action of 1198662on1198721015840 to 119866 by 120590 sdot 1198981015840 =

1198981015840 forall(1205901198981015840) isin 119866

1times 1198721015840 We easily see that the application

120595 119872 rarr 1198721015840 is an isomorphism of 119877[119866]-modules

Let 119870 be a commutative field of characteristic 119901 gt 0 Let119866 = 119866

1times 1198662 where 119866

1= 119862119901119898 and 119866

2= 119862119901119899 are two cyclic

groups of respective orders 119901119898 and 119901119899 and are generatedrespectively by 120590

1and 120590

2 and let 119869

1be the Jacobson radical

of 119870[1198661]

Proposition 15 Let119872 be a finitely generated 119870[119866]-moduleIf 1198691119872 = 0 then there exists a nonzero natural number 1198991015840 such

that119872 cong oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 119901119899 as119870[119866]-modules

where the action of 1198661on oplus119899

1015840

119894=1(1205902minus 1)119896119894119870[1198662] is trivial

Proof If 1198691119872 = 0 then the action of 119866

1on119872 is trivial since

1198691= (1205901minus 1)119870[119866

1] By Proposition 13 there exists a nonzero

natural number 1198991015840 such that 119872 cong oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662]

0 le 119896119894lt 119901119899 as 119870[119866

2]-modules Then Lemma 14 allows

concluding the following

Algebra 5

Theorem 16 Let 119872 be a finitely generated 119870[119866]-module If1198692

1119872 = 0 then there exist two nonzero natural numbers 1198991015840 and

11989910158401015840 and two119870[119866]-modules119872

1= oplus1198991015840

119894=1(1205902minus1)119896119894119870[1198662] 0 le 119896

119894lt

119901119899 and 119872

2= oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] 0 le 119896

1015840

119894lt 119901119899 where the

action of 1198661on1198721and119872

2is trivial and there is a derivation

119865 from 119870[119866] in119867119900119898119870(11987221198721) such that119872 cong 119872

1times1198651198722

Proof We have the exact sequence of 119870[119866]-modules 119874 rarr

1198691119872 997893rarr 119872 rarr 119872119869

1119872 rarr 119874 As 1198692

1119872 = 0 119869

1(1198691119872) = 0

So by Proposition 15 there exists a nonzero natural number1198991015840 such that 119869

1119872 cong oplus

1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 119901119899 as

119870[119866]-modules where the action of1198661on oplus119899

1015840

119894=1(1205902minus1)119896119894119870[1198662]

is trivial We set 1198721

= oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] We have

1198691(1198721198691119872) = 0 So by Proposition 15 there exists a nonzero

natural number 11989910158401015840 such that1198721198691119872 cong oplus

11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662]

0 le 1198961015840

119894lt 119901119899 as 119870[119866]-modules where the action of 119866

1

on oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] is trivial We set 119872

2= oplus11989910158401015840

119894=1(1205902minus

1)1198961015840

119894119870[1198662] Then Proposition 12 shows that 119872 cong 119872

1times1198651198722

for a derivation 119865 from119870[119866] in Hom119870(11987221198721)

If 119901 = 2 1198661= 1198622 and 119866

2= 1198622119899 then we have the

following corollary

Corollary 17 For all finitely generated 119870[119866]-modules thereexist two nonzero natural numbers 1198991015840 and 11989910158401015840 and two 119870[119866]-modules 119872

1= oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 2119899 and

1198722= oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] 0 le 119896

1015840

119894lt 2119899 where the action

of1198661on1198721and119872

2is trivial and there is a derivation 119865 from

119870[119866] in119867119900119898119870(11987221198721) such that119872 cong 119872

1times1198651198722

Proof We have 1198691= (1205901minus 1)119870[119866

1] and as (120590

1minus 1)2= 0 since

the field 119870 is of characteristic 119901 = 2 11986921= 0 So 1198692

1119872 = 0 for

119870[119866]-module of finite type119872The rest is a simple applicationof Theorem 16

Now we return to cases 1198661= 119862119901119898 and 119866

2= 119862119901119899 where

1198662is generated by an element 120590

2 and let 119869

2be the Jacobson

radical of 119870[1198662] For an integer 119896 = 119901

119903 with 0 lt 119903 le 119899 andfor the subgroup 119867

2of 1198662generated by 120590119901

119899minus119903

2 we have the

following result

Theorem 18 Let119872 be a finitely generated119870[119866]-module with119869119896+1

2119872 = 0 If 119872119869

119896

2119872darr1198661

is 119877-free and of type 119896 then thereexist two nonzero natural numbers 1198991015840 and 11989910158401015840 and two 119870[119866]-modules119872

1= oplus1198991015840

119894=1(1205901minus 1)119896119894119870[1198661] 0 le 119896

119894lt 119901119899 and119872

2=

(119870[1198661times 1198672])11989910158401015840

where the action of 1198662on1198721is trivial and

there is a derivation 119865 from119870[119866] in119867119900119898119870(11987221198721) such that

119872 cong 1198721times1198651198722

Proof We have the following exact sequence

119874 997888rarr 119869119896

2119872 997893rarr 119872 997888rarr

119872

119869119896

2119872

997888rarr 119874 (13)

As 1198692(119869119896

2119872) = 0 by Proposition 15 there exists a nonzero

natural number 1198991015840 such that 1198691198962119872 ≃ oplus

1198991015840

119894=1(1205901minus 1)119896119894119870[1198661]

0 le 119896119894lt 119901119899 as 119870[119866]-modules where the action of 119866

2on

oplus1198991015840

119894=1(1205901minus 1)119896119894119870[1198661] is trivial 119872119869

119896

2119872 is a 119870[119866]-module of

type 119896 with 1198691198962(119872119869119896

2119872) = 0 more119872119869

119896

2119872darr1198661

is 119877-free and119896 = 119901

119903 with 0 lt 119903 le 119899 Then Theorem 9 shows that119872119869119896

2119872

is a free119870[1198661times1198672]-moduleTherefore there exists a nonzero

natural number 11989910158401015840 such that119872119869119896

2119872 ≃ (119870[119866

1times1198672])11989910158401015840

Therest is a simple application of Proposition 12

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S O Kaptanoglu ldquoStructure and detection theorems fork[C2timesC4]-modulesrdquo The Rendiconti del Seminario Matematico

della Universita di Padova vol 123 pp 169ndash189 2010[2] P Webb Finite Group Representations for the Pure Mathemati-

cian University of Minnesota 2013[3] J Thevenaz Representations lineaires de groupes finis en car-

acteristique p119903 Universite de Geneve 1980[4] K W Gruenberg Cohomological Topics in Group Theory vol

143 of Lecture Notes in Mathematics Springer Berlin Germany1970

[5] G Karpilovsky The Jacobson Radical of Group Algebras NotasDeMathematica vol 135 Elsevier Science NewYork NY USA1987

[6] J Thevenaz ldquoRepresentations of finite groups in characteristic119901119903rdquo Journal of Algebra vol 72 no 2 pp 478ndash500 1981

[7] C W Curtis and I Reiner Methods of Representation TheoryWiley-Interscience New York NY USA 1981

[8] M E Charkani and S Bouhamidi ldquoModular representationsof Loewy length twordquo International Journal of Mathematics andMathematical Sciences no 70 pp 4399ndash4408 2003

[9] B Huppert and N Blackburn Finite Groups II Springer BerlinGermany 1982

Submit your manuscripts athttpwwwhindawicom

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

2 Algebra

119872 an 119877[119866]-module (not necessarily finitely generated) and119873a submodule of119872 and 120590 isin 119866 Then one has the following

(1) if (120590 minus 1)119872 = 119872 then119872 = 0

(2) if119872 = (120590 minus 1)119872 +119873 then119872 = 119873

(3) if 119909119894 119894 isin 119868 are representatives in 119872 of a generating

family of119872(120590 minus 1)119872 then (119909119894)119894isin119868

generate119872

Proof (1) Let 119901119903 be the order of 119866 We have

(120590 minus 1)119901119903

=

119901119903

sum

119894=0

119862119894

119901119903 (minus1)

119894120590119901119903minus119894

= 120590119901119903

+ (minus1)119901119903

+

119901119903minus1

sum

119894=1

119862119894

119901119903 (minus1)

119894120590119901119903minus119894

=

119901119903minus1

sum

119894=1

119862119894

119901119903 (minus1)

119894120590119901119903minus119894

= 119901

119901119903minus1

sum

119894=1

1

119901119862119894

119901119903(minus1 )

119894120590119901119903minus119894

(1)

For 1 le 119894 le 119901119903 minus 1 119901 | 119862119894119901119903 so (1119901)119862119894

119901119903 is a natural number

So

(120590 minus 1)119896119901119903

= ((120590 minus 1)119901119903

)119896

= (119901

119901119903minus1

sum

119894=1

1

119901119862119894

119901119903 (minus1)

119894120590119901119903minus119894)

119896

= 119901119896(

119901119903minus1

sum

119894=1

1

119901119862119894

119901119903 (minus1)

119894120590119901119903minus119894)

119896

= 0

(2)

since 119877 has characteristic 119901119896 Now 119872 = (120590 minus 1)119872 = (120590 minus

1)2119872 = sdot sdot sdot = (120590 minus 1)

119896119901119903

119872 = 0(2) If119872 = (120590minus1)119872+119873 then119872119873 = ((120590minus1)119872+119873)119873 =

(120590 minus 1)(119872119873) and then by (1)119872119873 = 0 and then119872 = 119873(3) If119873 is the submodule generated by 119909

119894 then119872 = (120590 minus

1)119872 +119873 and then by (2) we have119872 = 119873

Remark 2 Lemma 1 remains true if 119901 = 2 and 119877 is ofcharacteristic 2

For a ring 119877 of prime characteristic 119901 and for a cyclicgroup 119866 of order 119901119903 generated by an element 120590 we have thefollowing lemma

Lemma 3 Let 119896 = 1199011199031015840

with 0 lt 1199031015840 le 119903 and let119867 be a subgroupof 119866 generated by 120590119901

119903119896 Then one has 119877[119866](120590 minus 1)119896119877[119866] cong

119877[119867] (as 119877-algebras)

Proof Define

120595 119877 [119866]

(120590 minus 1)119896119877 [119866]

997888rarr 119877 [119867]

120590 997891997888rarr 120590119901119903119896

(3)

where120595 is awell-defined119877-algebra homomorphism It is easyto see that 120595 is surjective As 119877[119866](120590 minus 1)119896119877[119866] and 119877[119867]are finite free modules of the same rank 119896 over 119877 120595 is anisomorphism

Remark 4 With the notation of Lemma 3 119867 is simply thesubgroup of 119866 generated by 120590119901

119903minus1199031015840

Let 119870 be a commutative field of characteristic 119901 gt 0 andlet 119866 be a direct product of two finite groups 119866

1and 119866

2 We

have 119870[119866] = 119870[1198661times 1198662] cong 119877[119866

2] where 119877 = 119870[119866

1]

Assume that 1198661is a cyclic group of order 119901119898 generated

by 1205901and 119901 does not divide the order of 119866

2 119877 = 119870[119866

1] is a

principal Artinian local ring Indeed 119870[1198661] cong 119870[119883](119883 minus

1)119901119898

this isomorphism is induced by the homomorphismΨ 119870[119883] rarr 119870[119866] defined by Ψ(119883) = 120590

1 119870[119883](119883 minus 1)

119901119898

is a principal Artinian local ring with residue field 119870 (up toisomorphism) whosemaximal ideal is generated by119883 minus 1 So119877 is a principal Artinian local ring with residue field119870 (up toisomorphism) whose maximal ideal is generated by 120590

1minus 1

We have119870[119866] cong 119877[1198662] where 119877 is a principal Artinian local

ring of residue field119870The characteristic of119870 does not dividethe order of 119866

2 Under these conditions we can apply [3

Theorem 36] to have a complete system of indecomposablepairwise nonisomorphic119870[119866]-modules

In the remainder of this section we assume that1198661= 119862119901119898

and 1198662= 119862119901119899 are two cyclic groups of respective orders 119901119898

and 119901119899 and are generated respectively by 1205901and 1205902 We have

119870[119866] cong 119877[1198662] As 119877 is a commutative ring and local and

1198662is a 119901-group by [4 Proposition 10 page 239] 119877[119866

2] is a

local ringTherefore119870[119866] is a local ring As119870 is commutativering and local and 119866 is a 119901-group 119870[119866] is a local ring by [4Proposition 10 page 239] So the 119870[119866]-projective modulesare free 119870[119866]-modules

Lemma 5 Let119872 be a119870[119866]-module Then119872(1205901minus 1)119872 is a

119870[1198662]-module (also119872(120590

2minus 1)119872 is a 119870[119866

1]-module)

Proof This lemma is a particular case of a more generalresult (see [5 page 386]) But for this particular case wecan give the following direct proof 119872(120590

1minus 1)119872 is a

(119870[1198661](1205901minus1)119870[119866

1])[1198662]-module andwehave already seen

that (1205901minus 1)119870[119866

1] is the unique maximal ideal of119870[119866

1] and

119870[1198661](1205901minus 1)119870[119866

1] cong 119870 So 119872(120590

1minus 1)119872 is a 119870[119866

2]-

moduleSimilarly we show that119872(120590

2minus 1)119872 is a119870[119866

1]-module

Proposition 6 Let119872 be a free 119870[119866]-module of rank 119897 Then119872(120590

1minus 1)119872 is a free 119870[119866

2]-module and119872(120590

2minus 1)119872 is a

free 119870[1198661]-module of the same rank 119897

Algebra 3

Proof As 119870[119866] is a local ring 119870[119866]-projective modules arefree 119870[119866]-modules and therefore this proposition is onlya particular case of a more general result (see [5 Lemma22]) But for this particular case we can give the followingspecific proof we have 119872 cong (119870[119866])

119897= (119870[119866

1times 1198662])119897 So

119872 cong (119870[1198661][1198662])119897 Then we have

(1205901minus 1)119872 cong (120590

1minus 1) (119870 [119866

1][1198662])119897

cong ((1205901minus 1)119870 [119866

1][1198662])119897

(4)

Hence

119872

(1205901minus 1)119872

cong(119870 [119866

1][1198662])119897

((1205901minus 1)119870 [119866

1][1198662])119897

cong (119870 [1198661][1198662]

(1205901minus 1)119870 [119866

1][1198662])

119897

cong (119870 [1198661]

(1205901minus 1)119870 [119866

1][1198662])

119897

(5)

As 119870[1198661](1205901minus 1)119870[119866

1] cong 119870 (as we have already seen)

119872(1205901minus 1)119872 cong (119870[119866

2])119897 So119872(120590

1minus 1)119872 is a free 119870[119866

2]-

module of rank 119897Similarly we show that 119872(120590

2minus 1)119872 is a free 119870[119866

1]-

module of rank 119897

Proposition 7 Let 119872 be a 119870[119866]-module If 119872darr1198661

is a free119870[1198661]-module and119872(120590

1minus1)119872 is a free119870[119866

2]-module then

119872 is a free 119870[119866]-module

Proof 119877 = 119870[1198661] is a principal Artinian local ring with

residue field 119870 and 1205901minus 1 is a generator of its maximal ideal

119872darr1198661

is a free119877-module and119876 = 119872(1205901minus1)119872 is a projective

119870[1198662]-submodule of 119872(120590

1minus 1)119872 Then 119872 = 119875 oplus 119872

1015840where 119875 is a projective 119877[119866

2]-module and 119875(120590

1minus1)119875 cong 119876 =

119872(1205901minus 1)119872 (according to [3 Proposition 413]) We have

119872

(1205901minus 1)119872

cong119875 oplus119872

1015840

(1205901minus 1) (119875 oplus1198721015840)

cong119875 oplus119872

1015840

(1205901minus 1) 119875 oplus (120590

1minus 1)1198721015840

cong119875

(1205901minus 1) 119875

oplus1198721015840

(1205901minus 1)1198721015840

cong119872

(1205901minus 1)119872

oplus1198721015840

(1205901minus 1)1198721015840

(6)

So1198721015840(1205901minus1)119872

1015840= 0 By Nakayamarsquos lemma and the remark

following it 1198721015840 = 0 Therefore 119872 = 119875 which is projective119877[1198662]-module As 119877[119866

2] cong 119870[119866] is a local ring119872 is a free

119870[119866]-module

Let 119869119894be the Jacobson radical of119870[119866

119894] for 119894 isin 1 2 Note

that if 119870 is of characteristic 119901 (as here) and 1198661015840 is a cyclic 119901-group then the Jacobson radical of 119870[1198661015840] is none other than(120590 minus 1)119870[119866

1015840] where 120590 is a generator of 1198661015840 (see [5 page 122])

Let 119872 be a finitely generated 119870[119866]-module and 119896 anatural number such that 1 le 119896 le 119901119899 As119870[119866] cong 119877[119866

2]119872 is

a 119877[1198662]-module So119872119869

1119872 is a119870[119883](119883 minus 1)119901

119899

-module119872is called of type 119896 if119872119869

1119872 is a free 119870[119883](119883 minus 1)

119896-module(terminology of [6])

Lemma 8 If119872 is a 119870[119866]-module of type 119896 with 119896 = 119901119903 and0 lt 119903 le 119899 and119867

2is the subgroup of119866

2generated by 120590119901

119899minus119903

2 then

1198721198691119872 is a free 119870[119867

2]-module

Proof As 119872 is of type 119896 1198721198691119872 is a free 119870[119883](119883 minus 1)

119896-module Define

120595 119870 [119883]

(119883 minus 1)119896997888rarr 119870[119867

2]

119883 997891997888rarr 120590119901119899minus119903

2

(7)

where120595 is a well-defined119870-algebra homomorphism It is notdifficult to show that120595 is an isomorphism (using an argumentsimilar to that done in the proof of Lemma 3) So119872119869

1119872 is

a free119870[1198672]-module

Theorem 9 Let119872 be a 119870[119866]-module of type 119896 with 1198691198962119872 =

0 and let 1198672be the subgroup of 119866

2generated by 120590119901

119899minus119903

2with

0 lt 119903 le 119899 If 119872darr1198661

is 119877-free and 119896 = 119901119903 then 119872 is a free

119870[1198661times 1198672]-module

Proof 119872 is an 119877[1198662]-module 119877-free We have 119869119896

2119872 = 0 so

(1205902minus 1)119896119872 = 0 and therefore ((120590

2minus 1)119896119877[1198662])119872 = 0 So

119872 is an 119877[1198662](1205902minus 1)119896119877[1198662]-module 119877-free By Lemma 3

119877[1198662](1205902minus 1)119896119877[1198662] cong 119877[119867

2] then 119872 is an 119877[119867

2]-

module 119877-free1198721198691119872 is a free 119870[119883](119883 minus 1)

119896-module soby Lemma 8 this is a free119870[119867

2]-module In conclusion119872 is

a 119870[1198661times 1198672]-module such that

119872darr1198661

is a free 119896 [1198661]-module

119872

1198691119872

is a free 119870[1198672]-module

(8)

So by Proposition 7 119872 is a free119870[1198661times 1198672]-module

In Theorem 9 we assumed that the 119896[119866]-module 119872

satisfies the following condition119872darr1198661

is119877-free So it is usefulto know when this condition is satisfiedThis is the subject ofthe following result

Theorem 10 Let119872 be a 119896[119866]-module and 120590 an element of 119866of order 119901119903 The following conditions are equivalent

(1) 119872darr⟨120590⟩

is free(2) dim

119896(119872) = (119901

119903(119901119903minus 1))dim

119896((120590 minus 1)119872)

(3) dim119896(119872) = dim

119896((120590 minus 1)119872) + dim

119896((120590 minus 1)

119901119903minus1119872)

Proof (1) rArr (2) Assume that 119872darr⟨120590⟩

is free There exists anonzero natural number 119899 such that119872darr

⟨120590⟩cong (119896[⟨120590⟩])

119899 Theendomorphism 120593 of 119872 defined by 120593(119898) = (120590 minus 1)119898 for all

4 Algebra

119898 isin 119872 is nilpotent of nilpotency index 119901119903 and 119896[⟨120590⟩] isan indecomposable 119896[⟨120590⟩]-module Therefore119872 has a basisin which the matrix of 120593 is a Jordan matrix This matrix isformed of 119899 blocks of order 119901119903 all equal to

(

0 1

d dd 1

0

) (9)

So dim119896(119872) = 119901

119903times119899We can easily see that dim

119896((120590minus1)119872) =

119899times(119901119903minus1)Therefore dim

119896(119872) = (119901

119903(119901119903minus1))dim

119896((120590minus1)119872)

(2) rArr (3) Now assume that dim119896(119872) = (119901

119903(119901119903minus

1))dim119896((120590minus1)119872) So dim

119896(119872) = 119901

119903times(dim

119896(119872)minusdim

119896((120590minus

1)119872)) As dim119896(119872)minusdim

119896((120590minus1)119872) is the number of blocks

of the Jordan matrix of 120593 the order of each block is less thanor equal to 119901119903 and dim

119896(119872) is equal to the sum of the orders

of these blocks then the order of each block is 119901119903 Thereforedim119896((120590 minus 1)

119901119903minus1119872) is equal to the number of Jordan blocks

of 120593 So dim119896(119872) minus dim

119896((120590 minus 1)119872) = dim

119896((120590 minus 1)

119901119903minus1119872)

that is dim119896(119872) = dim

119896((120590 minus 1)119872) + dim

119896((120590 minus 1)

119901119903minus1119872)

(3) rArr (1) Assume that dim119896(119872) = dim

119896((120590 minus 1)119872) +

dim119896((120590 minus 1)

119901119903minus1119872) So dim

119896((120590 minus 1)

119901119903minus1119872) is equal to the

number of Jordan blocks of 120593 Therefore the order of eachJordan block is equal to 119901119903 So the modules contained in adecomposition of119872darr

⟨120590⟩as a direct sum of indecomposable

modules are of the form 119896[⟨120590⟩] that is119872darr⟨120590⟩

is free

3 Classification of FinitelyGenerated 119870[119862

119901119898 times 119862

119901119899]-Modules

Use of Module Extensions

Let 119866 be a finite group and let 119877 be a ring Let 1198721and 119872

2

be two 119877[119866]-modules We put 119879 = Hom119877(11987221198721) 119879 has a

natural structure as a (119877[119866] 119877[119866])-bimodule centralized by119877 (see [7 section 25]) Explicitly we have

(120582119891)1198982= 120582 sdot 119891 (119898

2) (119891120582)119898

2= 119891 (120582119898

2)

forall120582 isin 119877 [119866] 1198982isin 1198722 119891 isin 119879

(10)

A derivation 119865 119877[119866] rarr 119879 is an 119877-homomorphismsatisfying

1198651205821015840120582= 1205821015840119865120582+ 1198651205821015840120582 forall120582 120582

1015840isin 119877 [119866] (11)

Derivations from 119877[119866] into 119879 form an 119877-moduleDer(119877[119866] 119879) For 119865 isin Der(119877[119866] 119879) we equip119872

1times1198722with

an 119877[119866]-module structure by

120582 (1198981 1198982) = (120582119898

1+ 119865120582(1198982) 120582119898

2)

forall (1198981 1198982) isin 119872

1times1198722 120582 isin 119877 [119866]

(12)

This 119877[119866]-module is denoted by1198721times1198651198721as in [8]

An extension of 1198722by 119872

1is an 119877[119866]-exact sequence

119874 rarr 1198721rarr 119883 rarr 119872

2rarr 119874 Let 119874 rarr 119872

1

119906

997888rarr

119883V997888rarr 119872

2rarr 119874 and 119874 rarr 119872

1

1199061015840

997888rarr 1198831015840

V1015840997888rarr 119872

2rarr 119874

be a pair of extensions of1198722by1198721These two extensions are

equivalent if there exists an isomorphism of 119877[119866]-modulesΦ 119883 rarr 119883

1015840 such that Φ119900119906 = 1199061015840 and V1015840119900Φ = V

These equivalence classes of extensions form an 119877-moduleExt1119877[119866]

(11987221198721) The 119877[119866]-modules sequence 119874 rarr 119872

1

119894

997888rarr

1198721times1198651198722

119895

997888rarr 1198722rarr 119874 where 119894 and 119895 denote respectively

the canonical injection from1198721to1198721times1198651198722and the second

projection from 1198721times1198651198722to 1198722 is exact The equivalence

class of this sequence is denoted by [1198721times1198651198722]

Remark 11 With the previous notations derivations 119865 andmodules 119872

1times1198651198722play the same role as the cocycles 120572 and

modules1198721times1205721198722defined in [8]

FromProposition 2510 of [7]wehave the following result

Proposition 12 The correspondence 120579 Der(119877[119866] 119879) rarr

1198641199091199051

119877[119866](11987221198721) defined by 120579(119865) = [119872

1times1198651198722] is surjective

whenever1198722is finitely generated and projective as 119877-module

From Theorems 52 and 53 of [9] we have the followingresult

Proposition 13 Let 119866 be a cyclic group of order 119901119903 generatedby an element 120590 119870 a field of characteristic 119901 and 119872 anindecomposable 119870[119866]-module Then119872 is isomorphic to (120590 minus1)119904119870[119866] where 119904 is a natural number strictly less than 119901119903

Lemma 14 Let 119877 be a ring and 119866 = 1198661times 1198662a direct product

of two finite groups Let 119872 be an 119877[119866]-module such that theaction of 119866

1on119872 is trivial and let1198721015840 be an 119877[119866

2]-module If

119872 is isomorphic to1198721015840 as 119877[1198662]-modules and if we extend the

action of 1198662on1198721015840 to 119866 by 120590 sdot 1198981015840 = 1198981015840 forall(1205901198981015840) isin 119866

1times 1198721015840

then119872 is isomorphic to1198721015840 as 119877[119866]-modules

Proof Let 120595 119872 rarr 1198721015840 be an isomorphism of 119877[119866

2]-

modules We extend the action of 1198662on1198721015840 to 119866 by 120590 sdot 1198981015840 =

1198981015840 forall(1205901198981015840) isin 119866

1times 1198721015840 We easily see that the application

120595 119872 rarr 1198721015840 is an isomorphism of 119877[119866]-modules

Let 119870 be a commutative field of characteristic 119901 gt 0 Let119866 = 119866

1times 1198662 where 119866

1= 119862119901119898 and 119866

2= 119862119901119899 are two cyclic

groups of respective orders 119901119898 and 119901119899 and are generatedrespectively by 120590

1and 120590

2 and let 119869

1be the Jacobson radical

of 119870[1198661]

Proposition 15 Let119872 be a finitely generated 119870[119866]-moduleIf 1198691119872 = 0 then there exists a nonzero natural number 1198991015840 such

that119872 cong oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 119901119899 as119870[119866]-modules

where the action of 1198661on oplus119899

1015840

119894=1(1205902minus 1)119896119894119870[1198662] is trivial

Proof If 1198691119872 = 0 then the action of 119866

1on119872 is trivial since

1198691= (1205901minus 1)119870[119866

1] By Proposition 13 there exists a nonzero

natural number 1198991015840 such that 119872 cong oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662]

0 le 119896119894lt 119901119899 as 119870[119866

2]-modules Then Lemma 14 allows

concluding the following

Algebra 5

Theorem 16 Let 119872 be a finitely generated 119870[119866]-module If1198692

1119872 = 0 then there exist two nonzero natural numbers 1198991015840 and

11989910158401015840 and two119870[119866]-modules119872

1= oplus1198991015840

119894=1(1205902minus1)119896119894119870[1198662] 0 le 119896

119894lt

119901119899 and 119872

2= oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] 0 le 119896

1015840

119894lt 119901119899 where the

action of 1198661on1198721and119872

2is trivial and there is a derivation

119865 from 119870[119866] in119867119900119898119870(11987221198721) such that119872 cong 119872

1times1198651198722

Proof We have the exact sequence of 119870[119866]-modules 119874 rarr

1198691119872 997893rarr 119872 rarr 119872119869

1119872 rarr 119874 As 1198692

1119872 = 0 119869

1(1198691119872) = 0

So by Proposition 15 there exists a nonzero natural number1198991015840 such that 119869

1119872 cong oplus

1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 119901119899 as

119870[119866]-modules where the action of1198661on oplus119899

1015840

119894=1(1205902minus1)119896119894119870[1198662]

is trivial We set 1198721

= oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] We have

1198691(1198721198691119872) = 0 So by Proposition 15 there exists a nonzero

natural number 11989910158401015840 such that1198721198691119872 cong oplus

11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662]

0 le 1198961015840

119894lt 119901119899 as 119870[119866]-modules where the action of 119866

1

on oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] is trivial We set 119872

2= oplus11989910158401015840

119894=1(1205902minus

1)1198961015840

119894119870[1198662] Then Proposition 12 shows that 119872 cong 119872

1times1198651198722

for a derivation 119865 from119870[119866] in Hom119870(11987221198721)

If 119901 = 2 1198661= 1198622 and 119866

2= 1198622119899 then we have the

following corollary

Corollary 17 For all finitely generated 119870[119866]-modules thereexist two nonzero natural numbers 1198991015840 and 11989910158401015840 and two 119870[119866]-modules 119872

1= oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 2119899 and

1198722= oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] 0 le 119896

1015840

119894lt 2119899 where the action

of1198661on1198721and119872

2is trivial and there is a derivation 119865 from

119870[119866] in119867119900119898119870(11987221198721) such that119872 cong 119872

1times1198651198722

Proof We have 1198691= (1205901minus 1)119870[119866

1] and as (120590

1minus 1)2= 0 since

the field 119870 is of characteristic 119901 = 2 11986921= 0 So 1198692

1119872 = 0 for

119870[119866]-module of finite type119872The rest is a simple applicationof Theorem 16

Now we return to cases 1198661= 119862119901119898 and 119866

2= 119862119901119899 where

1198662is generated by an element 120590

2 and let 119869

2be the Jacobson

radical of 119870[1198662] For an integer 119896 = 119901

119903 with 0 lt 119903 le 119899 andfor the subgroup 119867

2of 1198662generated by 120590119901

119899minus119903

2 we have the

following result

Theorem 18 Let119872 be a finitely generated119870[119866]-module with119869119896+1

2119872 = 0 If 119872119869

119896

2119872darr1198661

is 119877-free and of type 119896 then thereexist two nonzero natural numbers 1198991015840 and 11989910158401015840 and two 119870[119866]-modules119872

1= oplus1198991015840

119894=1(1205901minus 1)119896119894119870[1198661] 0 le 119896

119894lt 119901119899 and119872

2=

(119870[1198661times 1198672])11989910158401015840

where the action of 1198662on1198721is trivial and

there is a derivation 119865 from119870[119866] in119867119900119898119870(11987221198721) such that

119872 cong 1198721times1198651198722

Proof We have the following exact sequence

119874 997888rarr 119869119896

2119872 997893rarr 119872 997888rarr

119872

119869119896

2119872

997888rarr 119874 (13)

As 1198692(119869119896

2119872) = 0 by Proposition 15 there exists a nonzero

natural number 1198991015840 such that 1198691198962119872 ≃ oplus

1198991015840

119894=1(1205901minus 1)119896119894119870[1198661]

0 le 119896119894lt 119901119899 as 119870[119866]-modules where the action of 119866

2on

oplus1198991015840

119894=1(1205901minus 1)119896119894119870[1198661] is trivial 119872119869

119896

2119872 is a 119870[119866]-module of

type 119896 with 1198691198962(119872119869119896

2119872) = 0 more119872119869

119896

2119872darr1198661

is 119877-free and119896 = 119901

119903 with 0 lt 119903 le 119899 Then Theorem 9 shows that119872119869119896

2119872

is a free119870[1198661times1198672]-moduleTherefore there exists a nonzero

natural number 11989910158401015840 such that119872119869119896

2119872 ≃ (119870[119866

1times1198672])11989910158401015840

Therest is a simple application of Proposition 12

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S O Kaptanoglu ldquoStructure and detection theorems fork[C2timesC4]-modulesrdquo The Rendiconti del Seminario Matematico

della Universita di Padova vol 123 pp 169ndash189 2010[2] P Webb Finite Group Representations for the Pure Mathemati-

cian University of Minnesota 2013[3] J Thevenaz Representations lineaires de groupes finis en car-

acteristique p119903 Universite de Geneve 1980[4] K W Gruenberg Cohomological Topics in Group Theory vol

143 of Lecture Notes in Mathematics Springer Berlin Germany1970

[5] G Karpilovsky The Jacobson Radical of Group Algebras NotasDeMathematica vol 135 Elsevier Science NewYork NY USA1987

[6] J Thevenaz ldquoRepresentations of finite groups in characteristic119901119903rdquo Journal of Algebra vol 72 no 2 pp 478ndash500 1981

[7] C W Curtis and I Reiner Methods of Representation TheoryWiley-Interscience New York NY USA 1981

[8] M E Charkani and S Bouhamidi ldquoModular representationsof Loewy length twordquo International Journal of Mathematics andMathematical Sciences no 70 pp 4399ndash4408 2003

[9] B Huppert and N Blackburn Finite Groups II Springer BerlinGermany 1982

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Algebra 3

Proof As 119870[119866] is a local ring 119870[119866]-projective modules arefree 119870[119866]-modules and therefore this proposition is onlya particular case of a more general result (see [5 Lemma22]) But for this particular case we can give the followingspecific proof we have 119872 cong (119870[119866])

119897= (119870[119866

1times 1198662])119897 So

119872 cong (119870[1198661][1198662])119897 Then we have

(1205901minus 1)119872 cong (120590

1minus 1) (119870 [119866

1][1198662])119897

cong ((1205901minus 1)119870 [119866

1][1198662])119897

(4)

Hence

119872

(1205901minus 1)119872

cong(119870 [119866

1][1198662])119897

((1205901minus 1)119870 [119866

1][1198662])119897

cong (119870 [1198661][1198662]

(1205901minus 1)119870 [119866

1][1198662])

119897

cong (119870 [1198661]

(1205901minus 1)119870 [119866

1][1198662])

119897

(5)

As 119870[1198661](1205901minus 1)119870[119866

1] cong 119870 (as we have already seen)

119872(1205901minus 1)119872 cong (119870[119866

2])119897 So119872(120590

1minus 1)119872 is a free 119870[119866

2]-

module of rank 119897Similarly we show that 119872(120590

2minus 1)119872 is a free 119870[119866

1]-

module of rank 119897

Proposition 7 Let 119872 be a 119870[119866]-module If 119872darr1198661

is a free119870[1198661]-module and119872(120590

1minus1)119872 is a free119870[119866

2]-module then

119872 is a free 119870[119866]-module

Proof 119877 = 119870[1198661] is a principal Artinian local ring with

residue field 119870 and 1205901minus 1 is a generator of its maximal ideal

119872darr1198661

is a free119877-module and119876 = 119872(1205901minus1)119872 is a projective

119870[1198662]-submodule of 119872(120590

1minus 1)119872 Then 119872 = 119875 oplus 119872

1015840where 119875 is a projective 119877[119866

2]-module and 119875(120590

1minus1)119875 cong 119876 =

119872(1205901minus 1)119872 (according to [3 Proposition 413]) We have

119872

(1205901minus 1)119872

cong119875 oplus119872

1015840

(1205901minus 1) (119875 oplus1198721015840)

cong119875 oplus119872

1015840

(1205901minus 1) 119875 oplus (120590

1minus 1)1198721015840

cong119875

(1205901minus 1) 119875

oplus1198721015840

(1205901minus 1)1198721015840

cong119872

(1205901minus 1)119872

oplus1198721015840

(1205901minus 1)1198721015840

(6)

So1198721015840(1205901minus1)119872

1015840= 0 By Nakayamarsquos lemma and the remark

following it 1198721015840 = 0 Therefore 119872 = 119875 which is projective119877[1198662]-module As 119877[119866

2] cong 119870[119866] is a local ring119872 is a free

119870[119866]-module

Let 119869119894be the Jacobson radical of119870[119866

119894] for 119894 isin 1 2 Note

that if 119870 is of characteristic 119901 (as here) and 1198661015840 is a cyclic 119901-group then the Jacobson radical of 119870[1198661015840] is none other than(120590 minus 1)119870[119866

1015840] where 120590 is a generator of 1198661015840 (see [5 page 122])

Let 119872 be a finitely generated 119870[119866]-module and 119896 anatural number such that 1 le 119896 le 119901119899 As119870[119866] cong 119877[119866

2]119872 is

a 119877[1198662]-module So119872119869

1119872 is a119870[119883](119883 minus 1)119901

119899

-module119872is called of type 119896 if119872119869

1119872 is a free 119870[119883](119883 minus 1)

119896-module(terminology of [6])

Lemma 8 If119872 is a 119870[119866]-module of type 119896 with 119896 = 119901119903 and0 lt 119903 le 119899 and119867

2is the subgroup of119866

2generated by 120590119901

119899minus119903

2 then

1198721198691119872 is a free 119870[119867

2]-module

Proof As 119872 is of type 119896 1198721198691119872 is a free 119870[119883](119883 minus 1)

119896-module Define

120595 119870 [119883]

(119883 minus 1)119896997888rarr 119870[119867

2]

119883 997891997888rarr 120590119901119899minus119903

2

(7)

where120595 is a well-defined119870-algebra homomorphism It is notdifficult to show that120595 is an isomorphism (using an argumentsimilar to that done in the proof of Lemma 3) So119872119869

1119872 is

a free119870[1198672]-module

Theorem 9 Let119872 be a 119870[119866]-module of type 119896 with 1198691198962119872 =

0 and let 1198672be the subgroup of 119866

2generated by 120590119901

119899minus119903

2with

0 lt 119903 le 119899 If 119872darr1198661

is 119877-free and 119896 = 119901119903 then 119872 is a free

119870[1198661times 1198672]-module

Proof 119872 is an 119877[1198662]-module 119877-free We have 119869119896

2119872 = 0 so

(1205902minus 1)119896119872 = 0 and therefore ((120590

2minus 1)119896119877[1198662])119872 = 0 So

119872 is an 119877[1198662](1205902minus 1)119896119877[1198662]-module 119877-free By Lemma 3

119877[1198662](1205902minus 1)119896119877[1198662] cong 119877[119867

2] then 119872 is an 119877[119867

2]-

module 119877-free1198721198691119872 is a free 119870[119883](119883 minus 1)

119896-module soby Lemma 8 this is a free119870[119867

2]-module In conclusion119872 is

a 119870[1198661times 1198672]-module such that

119872darr1198661

is a free 119896 [1198661]-module

119872

1198691119872

is a free 119870[1198672]-module

(8)

So by Proposition 7 119872 is a free119870[1198661times 1198672]-module

In Theorem 9 we assumed that the 119896[119866]-module 119872

satisfies the following condition119872darr1198661

is119877-free So it is usefulto know when this condition is satisfiedThis is the subject ofthe following result

Theorem 10 Let119872 be a 119896[119866]-module and 120590 an element of 119866of order 119901119903 The following conditions are equivalent

(1) 119872darr⟨120590⟩

is free(2) dim

119896(119872) = (119901

119903(119901119903minus 1))dim

119896((120590 minus 1)119872)

(3) dim119896(119872) = dim

119896((120590 minus 1)119872) + dim

119896((120590 minus 1)

119901119903minus1119872)

Proof (1) rArr (2) Assume that 119872darr⟨120590⟩

is free There exists anonzero natural number 119899 such that119872darr

⟨120590⟩cong (119896[⟨120590⟩])

119899 Theendomorphism 120593 of 119872 defined by 120593(119898) = (120590 minus 1)119898 for all

4 Algebra

119898 isin 119872 is nilpotent of nilpotency index 119901119903 and 119896[⟨120590⟩] isan indecomposable 119896[⟨120590⟩]-module Therefore119872 has a basisin which the matrix of 120593 is a Jordan matrix This matrix isformed of 119899 blocks of order 119901119903 all equal to

(

0 1

d dd 1

0

) (9)

So dim119896(119872) = 119901

119903times119899We can easily see that dim

119896((120590minus1)119872) =

119899times(119901119903minus1)Therefore dim

119896(119872) = (119901

119903(119901119903minus1))dim

119896((120590minus1)119872)

(2) rArr (3) Now assume that dim119896(119872) = (119901

119903(119901119903minus

1))dim119896((120590minus1)119872) So dim

119896(119872) = 119901

119903times(dim

119896(119872)minusdim

119896((120590minus

1)119872)) As dim119896(119872)minusdim

119896((120590minus1)119872) is the number of blocks

of the Jordan matrix of 120593 the order of each block is less thanor equal to 119901119903 and dim

119896(119872) is equal to the sum of the orders

of these blocks then the order of each block is 119901119903 Thereforedim119896((120590 minus 1)

119901119903minus1119872) is equal to the number of Jordan blocks

of 120593 So dim119896(119872) minus dim

119896((120590 minus 1)119872) = dim

119896((120590 minus 1)

119901119903minus1119872)

that is dim119896(119872) = dim

119896((120590 minus 1)119872) + dim

119896((120590 minus 1)

119901119903minus1119872)

(3) rArr (1) Assume that dim119896(119872) = dim

119896((120590 minus 1)119872) +

dim119896((120590 minus 1)

119901119903minus1119872) So dim

119896((120590 minus 1)

119901119903minus1119872) is equal to the

number of Jordan blocks of 120593 Therefore the order of eachJordan block is equal to 119901119903 So the modules contained in adecomposition of119872darr

⟨120590⟩as a direct sum of indecomposable

modules are of the form 119896[⟨120590⟩] that is119872darr⟨120590⟩

is free

3 Classification of FinitelyGenerated 119870[119862

119901119898 times 119862

119901119899]-Modules

Use of Module Extensions

Let 119866 be a finite group and let 119877 be a ring Let 1198721and 119872

2

be two 119877[119866]-modules We put 119879 = Hom119877(11987221198721) 119879 has a

natural structure as a (119877[119866] 119877[119866])-bimodule centralized by119877 (see [7 section 25]) Explicitly we have

(120582119891)1198982= 120582 sdot 119891 (119898

2) (119891120582)119898

2= 119891 (120582119898

2)

forall120582 isin 119877 [119866] 1198982isin 1198722 119891 isin 119879

(10)

A derivation 119865 119877[119866] rarr 119879 is an 119877-homomorphismsatisfying

1198651205821015840120582= 1205821015840119865120582+ 1198651205821015840120582 forall120582 120582

1015840isin 119877 [119866] (11)

Derivations from 119877[119866] into 119879 form an 119877-moduleDer(119877[119866] 119879) For 119865 isin Der(119877[119866] 119879) we equip119872

1times1198722with

an 119877[119866]-module structure by

120582 (1198981 1198982) = (120582119898

1+ 119865120582(1198982) 120582119898

2)

forall (1198981 1198982) isin 119872

1times1198722 120582 isin 119877 [119866]

(12)

This 119877[119866]-module is denoted by1198721times1198651198721as in [8]

An extension of 1198722by 119872

1is an 119877[119866]-exact sequence

119874 rarr 1198721rarr 119883 rarr 119872

2rarr 119874 Let 119874 rarr 119872

1

119906

997888rarr

119883V997888rarr 119872

2rarr 119874 and 119874 rarr 119872

1

1199061015840

997888rarr 1198831015840

V1015840997888rarr 119872

2rarr 119874

be a pair of extensions of1198722by1198721These two extensions are

equivalent if there exists an isomorphism of 119877[119866]-modulesΦ 119883 rarr 119883

1015840 such that Φ119900119906 = 1199061015840 and V1015840119900Φ = V

These equivalence classes of extensions form an 119877-moduleExt1119877[119866]

(11987221198721) The 119877[119866]-modules sequence 119874 rarr 119872

1

119894

997888rarr

1198721times1198651198722

119895

997888rarr 1198722rarr 119874 where 119894 and 119895 denote respectively

the canonical injection from1198721to1198721times1198651198722and the second

projection from 1198721times1198651198722to 1198722 is exact The equivalence

class of this sequence is denoted by [1198721times1198651198722]

Remark 11 With the previous notations derivations 119865 andmodules 119872

1times1198651198722play the same role as the cocycles 120572 and

modules1198721times1205721198722defined in [8]

FromProposition 2510 of [7]wehave the following result

Proposition 12 The correspondence 120579 Der(119877[119866] 119879) rarr

1198641199091199051

119877[119866](11987221198721) defined by 120579(119865) = [119872

1times1198651198722] is surjective

whenever1198722is finitely generated and projective as 119877-module

From Theorems 52 and 53 of [9] we have the followingresult

Proposition 13 Let 119866 be a cyclic group of order 119901119903 generatedby an element 120590 119870 a field of characteristic 119901 and 119872 anindecomposable 119870[119866]-module Then119872 is isomorphic to (120590 minus1)119904119870[119866] where 119904 is a natural number strictly less than 119901119903

Lemma 14 Let 119877 be a ring and 119866 = 1198661times 1198662a direct product

of two finite groups Let 119872 be an 119877[119866]-module such that theaction of 119866

1on119872 is trivial and let1198721015840 be an 119877[119866

2]-module If

119872 is isomorphic to1198721015840 as 119877[1198662]-modules and if we extend the

action of 1198662on1198721015840 to 119866 by 120590 sdot 1198981015840 = 1198981015840 forall(1205901198981015840) isin 119866

1times 1198721015840

then119872 is isomorphic to1198721015840 as 119877[119866]-modules

Proof Let 120595 119872 rarr 1198721015840 be an isomorphism of 119877[119866

2]-

modules We extend the action of 1198662on1198721015840 to 119866 by 120590 sdot 1198981015840 =

1198981015840 forall(1205901198981015840) isin 119866

1times 1198721015840 We easily see that the application

120595 119872 rarr 1198721015840 is an isomorphism of 119877[119866]-modules

Let 119870 be a commutative field of characteristic 119901 gt 0 Let119866 = 119866

1times 1198662 where 119866

1= 119862119901119898 and 119866

2= 119862119901119899 are two cyclic

groups of respective orders 119901119898 and 119901119899 and are generatedrespectively by 120590

1and 120590

2 and let 119869

1be the Jacobson radical

of 119870[1198661]

Proposition 15 Let119872 be a finitely generated 119870[119866]-moduleIf 1198691119872 = 0 then there exists a nonzero natural number 1198991015840 such

that119872 cong oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 119901119899 as119870[119866]-modules

where the action of 1198661on oplus119899

1015840

119894=1(1205902minus 1)119896119894119870[1198662] is trivial

Proof If 1198691119872 = 0 then the action of 119866

1on119872 is trivial since

1198691= (1205901minus 1)119870[119866

1] By Proposition 13 there exists a nonzero

natural number 1198991015840 such that 119872 cong oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662]

0 le 119896119894lt 119901119899 as 119870[119866

2]-modules Then Lemma 14 allows

concluding the following

Algebra 5

Theorem 16 Let 119872 be a finitely generated 119870[119866]-module If1198692

1119872 = 0 then there exist two nonzero natural numbers 1198991015840 and

11989910158401015840 and two119870[119866]-modules119872

1= oplus1198991015840

119894=1(1205902minus1)119896119894119870[1198662] 0 le 119896

119894lt

119901119899 and 119872

2= oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] 0 le 119896

1015840

119894lt 119901119899 where the

action of 1198661on1198721and119872

2is trivial and there is a derivation

119865 from 119870[119866] in119867119900119898119870(11987221198721) such that119872 cong 119872

1times1198651198722

Proof We have the exact sequence of 119870[119866]-modules 119874 rarr

1198691119872 997893rarr 119872 rarr 119872119869

1119872 rarr 119874 As 1198692

1119872 = 0 119869

1(1198691119872) = 0

So by Proposition 15 there exists a nonzero natural number1198991015840 such that 119869

1119872 cong oplus

1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 119901119899 as

119870[119866]-modules where the action of1198661on oplus119899

1015840

119894=1(1205902minus1)119896119894119870[1198662]

is trivial We set 1198721

= oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] We have

1198691(1198721198691119872) = 0 So by Proposition 15 there exists a nonzero

natural number 11989910158401015840 such that1198721198691119872 cong oplus

11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662]

0 le 1198961015840

119894lt 119901119899 as 119870[119866]-modules where the action of 119866

1

on oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] is trivial We set 119872

2= oplus11989910158401015840

119894=1(1205902minus

1)1198961015840

119894119870[1198662] Then Proposition 12 shows that 119872 cong 119872

1times1198651198722

for a derivation 119865 from119870[119866] in Hom119870(11987221198721)

If 119901 = 2 1198661= 1198622 and 119866

2= 1198622119899 then we have the

following corollary

Corollary 17 For all finitely generated 119870[119866]-modules thereexist two nonzero natural numbers 1198991015840 and 11989910158401015840 and two 119870[119866]-modules 119872

1= oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 2119899 and

1198722= oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] 0 le 119896

1015840

119894lt 2119899 where the action

of1198661on1198721and119872

2is trivial and there is a derivation 119865 from

119870[119866] in119867119900119898119870(11987221198721) such that119872 cong 119872

1times1198651198722

Proof We have 1198691= (1205901minus 1)119870[119866

1] and as (120590

1minus 1)2= 0 since

the field 119870 is of characteristic 119901 = 2 11986921= 0 So 1198692

1119872 = 0 for

119870[119866]-module of finite type119872The rest is a simple applicationof Theorem 16

Now we return to cases 1198661= 119862119901119898 and 119866

2= 119862119901119899 where

1198662is generated by an element 120590

2 and let 119869

2be the Jacobson

radical of 119870[1198662] For an integer 119896 = 119901

119903 with 0 lt 119903 le 119899 andfor the subgroup 119867

2of 1198662generated by 120590119901

119899minus119903

2 we have the

following result

Theorem 18 Let119872 be a finitely generated119870[119866]-module with119869119896+1

2119872 = 0 If 119872119869

119896

2119872darr1198661

is 119877-free and of type 119896 then thereexist two nonzero natural numbers 1198991015840 and 11989910158401015840 and two 119870[119866]-modules119872

1= oplus1198991015840

119894=1(1205901minus 1)119896119894119870[1198661] 0 le 119896

119894lt 119901119899 and119872

2=

(119870[1198661times 1198672])11989910158401015840

where the action of 1198662on1198721is trivial and

there is a derivation 119865 from119870[119866] in119867119900119898119870(11987221198721) such that

119872 cong 1198721times1198651198722

Proof We have the following exact sequence

119874 997888rarr 119869119896

2119872 997893rarr 119872 997888rarr

119872

119869119896

2119872

997888rarr 119874 (13)

As 1198692(119869119896

2119872) = 0 by Proposition 15 there exists a nonzero

natural number 1198991015840 such that 1198691198962119872 ≃ oplus

1198991015840

119894=1(1205901minus 1)119896119894119870[1198661]

0 le 119896119894lt 119901119899 as 119870[119866]-modules where the action of 119866

2on

oplus1198991015840

119894=1(1205901minus 1)119896119894119870[1198661] is trivial 119872119869

119896

2119872 is a 119870[119866]-module of

type 119896 with 1198691198962(119872119869119896

2119872) = 0 more119872119869

119896

2119872darr1198661

is 119877-free and119896 = 119901

119903 with 0 lt 119903 le 119899 Then Theorem 9 shows that119872119869119896

2119872

is a free119870[1198661times1198672]-moduleTherefore there exists a nonzero

natural number 11989910158401015840 such that119872119869119896

2119872 ≃ (119870[119866

1times1198672])11989910158401015840

Therest is a simple application of Proposition 12

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S O Kaptanoglu ldquoStructure and detection theorems fork[C2timesC4]-modulesrdquo The Rendiconti del Seminario Matematico

della Universita di Padova vol 123 pp 169ndash189 2010[2] P Webb Finite Group Representations for the Pure Mathemati-

cian University of Minnesota 2013[3] J Thevenaz Representations lineaires de groupes finis en car-

acteristique p119903 Universite de Geneve 1980[4] K W Gruenberg Cohomological Topics in Group Theory vol

143 of Lecture Notes in Mathematics Springer Berlin Germany1970

[5] G Karpilovsky The Jacobson Radical of Group Algebras NotasDeMathematica vol 135 Elsevier Science NewYork NY USA1987

[6] J Thevenaz ldquoRepresentations of finite groups in characteristic119901119903rdquo Journal of Algebra vol 72 no 2 pp 478ndash500 1981

[7] C W Curtis and I Reiner Methods of Representation TheoryWiley-Interscience New York NY USA 1981

[8] M E Charkani and S Bouhamidi ldquoModular representationsof Loewy length twordquo International Journal of Mathematics andMathematical Sciences no 70 pp 4399ndash4408 2003

[9] B Huppert and N Blackburn Finite Groups II Springer BerlinGermany 1982

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Algebra

119898 isin 119872 is nilpotent of nilpotency index 119901119903 and 119896[⟨120590⟩] isan indecomposable 119896[⟨120590⟩]-module Therefore119872 has a basisin which the matrix of 120593 is a Jordan matrix This matrix isformed of 119899 blocks of order 119901119903 all equal to

(

0 1

d dd 1

0

) (9)

So dim119896(119872) = 119901

119903times119899We can easily see that dim

119896((120590minus1)119872) =

119899times(119901119903minus1)Therefore dim

119896(119872) = (119901

119903(119901119903minus1))dim

119896((120590minus1)119872)

(2) rArr (3) Now assume that dim119896(119872) = (119901

119903(119901119903minus

1))dim119896((120590minus1)119872) So dim

119896(119872) = 119901

119903times(dim

119896(119872)minusdim

119896((120590minus

1)119872)) As dim119896(119872)minusdim

119896((120590minus1)119872) is the number of blocks

of the Jordan matrix of 120593 the order of each block is less thanor equal to 119901119903 and dim

119896(119872) is equal to the sum of the orders

of these blocks then the order of each block is 119901119903 Thereforedim119896((120590 minus 1)

119901119903minus1119872) is equal to the number of Jordan blocks

of 120593 So dim119896(119872) minus dim

119896((120590 minus 1)119872) = dim

119896((120590 minus 1)

119901119903minus1119872)

that is dim119896(119872) = dim

119896((120590 minus 1)119872) + dim

119896((120590 minus 1)

119901119903minus1119872)

(3) rArr (1) Assume that dim119896(119872) = dim

119896((120590 minus 1)119872) +

dim119896((120590 minus 1)

119901119903minus1119872) So dim

119896((120590 minus 1)

119901119903minus1119872) is equal to the

number of Jordan blocks of 120593 Therefore the order of eachJordan block is equal to 119901119903 So the modules contained in adecomposition of119872darr

⟨120590⟩as a direct sum of indecomposable

modules are of the form 119896[⟨120590⟩] that is119872darr⟨120590⟩

is free

3 Classification of FinitelyGenerated 119870[119862

119901119898 times 119862

119901119899]-Modules

Use of Module Extensions

Let 119866 be a finite group and let 119877 be a ring Let 1198721and 119872

2

be two 119877[119866]-modules We put 119879 = Hom119877(11987221198721) 119879 has a

natural structure as a (119877[119866] 119877[119866])-bimodule centralized by119877 (see [7 section 25]) Explicitly we have

(120582119891)1198982= 120582 sdot 119891 (119898

2) (119891120582)119898

2= 119891 (120582119898

2)

forall120582 isin 119877 [119866] 1198982isin 1198722 119891 isin 119879

(10)

A derivation 119865 119877[119866] rarr 119879 is an 119877-homomorphismsatisfying

1198651205821015840120582= 1205821015840119865120582+ 1198651205821015840120582 forall120582 120582

1015840isin 119877 [119866] (11)

Derivations from 119877[119866] into 119879 form an 119877-moduleDer(119877[119866] 119879) For 119865 isin Der(119877[119866] 119879) we equip119872

1times1198722with

an 119877[119866]-module structure by

120582 (1198981 1198982) = (120582119898

1+ 119865120582(1198982) 120582119898

2)

forall (1198981 1198982) isin 119872

1times1198722 120582 isin 119877 [119866]

(12)

This 119877[119866]-module is denoted by1198721times1198651198721as in [8]

An extension of 1198722by 119872

1is an 119877[119866]-exact sequence

119874 rarr 1198721rarr 119883 rarr 119872

2rarr 119874 Let 119874 rarr 119872

1

119906

997888rarr

119883V997888rarr 119872

2rarr 119874 and 119874 rarr 119872

1

1199061015840

997888rarr 1198831015840

V1015840997888rarr 119872

2rarr 119874

be a pair of extensions of1198722by1198721These two extensions are

equivalent if there exists an isomorphism of 119877[119866]-modulesΦ 119883 rarr 119883

1015840 such that Φ119900119906 = 1199061015840 and V1015840119900Φ = V

These equivalence classes of extensions form an 119877-moduleExt1119877[119866]

(11987221198721) The 119877[119866]-modules sequence 119874 rarr 119872

1

119894

997888rarr

1198721times1198651198722

119895

997888rarr 1198722rarr 119874 where 119894 and 119895 denote respectively

the canonical injection from1198721to1198721times1198651198722and the second

projection from 1198721times1198651198722to 1198722 is exact The equivalence

class of this sequence is denoted by [1198721times1198651198722]

Remark 11 With the previous notations derivations 119865 andmodules 119872

1times1198651198722play the same role as the cocycles 120572 and

modules1198721times1205721198722defined in [8]

FromProposition 2510 of [7]wehave the following result

Proposition 12 The correspondence 120579 Der(119877[119866] 119879) rarr

1198641199091199051

119877[119866](11987221198721) defined by 120579(119865) = [119872

1times1198651198722] is surjective

whenever1198722is finitely generated and projective as 119877-module

From Theorems 52 and 53 of [9] we have the followingresult

Proposition 13 Let 119866 be a cyclic group of order 119901119903 generatedby an element 120590 119870 a field of characteristic 119901 and 119872 anindecomposable 119870[119866]-module Then119872 is isomorphic to (120590 minus1)119904119870[119866] where 119904 is a natural number strictly less than 119901119903

Lemma 14 Let 119877 be a ring and 119866 = 1198661times 1198662a direct product

of two finite groups Let 119872 be an 119877[119866]-module such that theaction of 119866

1on119872 is trivial and let1198721015840 be an 119877[119866

2]-module If

119872 is isomorphic to1198721015840 as 119877[1198662]-modules and if we extend the

action of 1198662on1198721015840 to 119866 by 120590 sdot 1198981015840 = 1198981015840 forall(1205901198981015840) isin 119866

1times 1198721015840

then119872 is isomorphic to1198721015840 as 119877[119866]-modules

Proof Let 120595 119872 rarr 1198721015840 be an isomorphism of 119877[119866

2]-

modules We extend the action of 1198662on1198721015840 to 119866 by 120590 sdot 1198981015840 =

1198981015840 forall(1205901198981015840) isin 119866

1times 1198721015840 We easily see that the application

120595 119872 rarr 1198721015840 is an isomorphism of 119877[119866]-modules

Let 119870 be a commutative field of characteristic 119901 gt 0 Let119866 = 119866

1times 1198662 where 119866

1= 119862119901119898 and 119866

2= 119862119901119899 are two cyclic

groups of respective orders 119901119898 and 119901119899 and are generatedrespectively by 120590

1and 120590

2 and let 119869

1be the Jacobson radical

of 119870[1198661]

Proposition 15 Let119872 be a finitely generated 119870[119866]-moduleIf 1198691119872 = 0 then there exists a nonzero natural number 1198991015840 such

that119872 cong oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 119901119899 as119870[119866]-modules

where the action of 1198661on oplus119899

1015840

119894=1(1205902minus 1)119896119894119870[1198662] is trivial

Proof If 1198691119872 = 0 then the action of 119866

1on119872 is trivial since

1198691= (1205901minus 1)119870[119866

1] By Proposition 13 there exists a nonzero

natural number 1198991015840 such that 119872 cong oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662]

0 le 119896119894lt 119901119899 as 119870[119866

2]-modules Then Lemma 14 allows

concluding the following

Algebra 5

Theorem 16 Let 119872 be a finitely generated 119870[119866]-module If1198692

1119872 = 0 then there exist two nonzero natural numbers 1198991015840 and

11989910158401015840 and two119870[119866]-modules119872

1= oplus1198991015840

119894=1(1205902minus1)119896119894119870[1198662] 0 le 119896

119894lt

119901119899 and 119872

2= oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] 0 le 119896

1015840

119894lt 119901119899 where the

action of 1198661on1198721and119872

2is trivial and there is a derivation

119865 from 119870[119866] in119867119900119898119870(11987221198721) such that119872 cong 119872

1times1198651198722

Proof We have the exact sequence of 119870[119866]-modules 119874 rarr

1198691119872 997893rarr 119872 rarr 119872119869

1119872 rarr 119874 As 1198692

1119872 = 0 119869

1(1198691119872) = 0

So by Proposition 15 there exists a nonzero natural number1198991015840 such that 119869

1119872 cong oplus

1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 119901119899 as

119870[119866]-modules where the action of1198661on oplus119899

1015840

119894=1(1205902minus1)119896119894119870[1198662]

is trivial We set 1198721

= oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] We have

1198691(1198721198691119872) = 0 So by Proposition 15 there exists a nonzero

natural number 11989910158401015840 such that1198721198691119872 cong oplus

11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662]

0 le 1198961015840

119894lt 119901119899 as 119870[119866]-modules where the action of 119866

1

on oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] is trivial We set 119872

2= oplus11989910158401015840

119894=1(1205902minus

1)1198961015840

119894119870[1198662] Then Proposition 12 shows that 119872 cong 119872

1times1198651198722

for a derivation 119865 from119870[119866] in Hom119870(11987221198721)

If 119901 = 2 1198661= 1198622 and 119866

2= 1198622119899 then we have the

following corollary

Corollary 17 For all finitely generated 119870[119866]-modules thereexist two nonzero natural numbers 1198991015840 and 11989910158401015840 and two 119870[119866]-modules 119872

1= oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 2119899 and

1198722= oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] 0 le 119896

1015840

119894lt 2119899 where the action

of1198661on1198721and119872

2is trivial and there is a derivation 119865 from

119870[119866] in119867119900119898119870(11987221198721) such that119872 cong 119872

1times1198651198722

Proof We have 1198691= (1205901minus 1)119870[119866

1] and as (120590

1minus 1)2= 0 since

the field 119870 is of characteristic 119901 = 2 11986921= 0 So 1198692

1119872 = 0 for

119870[119866]-module of finite type119872The rest is a simple applicationof Theorem 16

Now we return to cases 1198661= 119862119901119898 and 119866

2= 119862119901119899 where

1198662is generated by an element 120590

2 and let 119869

2be the Jacobson

radical of 119870[1198662] For an integer 119896 = 119901

119903 with 0 lt 119903 le 119899 andfor the subgroup 119867

2of 1198662generated by 120590119901

119899minus119903

2 we have the

following result

Theorem 18 Let119872 be a finitely generated119870[119866]-module with119869119896+1

2119872 = 0 If 119872119869

119896

2119872darr1198661

is 119877-free and of type 119896 then thereexist two nonzero natural numbers 1198991015840 and 11989910158401015840 and two 119870[119866]-modules119872

1= oplus1198991015840

119894=1(1205901minus 1)119896119894119870[1198661] 0 le 119896

119894lt 119901119899 and119872

2=

(119870[1198661times 1198672])11989910158401015840

where the action of 1198662on1198721is trivial and

there is a derivation 119865 from119870[119866] in119867119900119898119870(11987221198721) such that

119872 cong 1198721times1198651198722

Proof We have the following exact sequence

119874 997888rarr 119869119896

2119872 997893rarr 119872 997888rarr

119872

119869119896

2119872

997888rarr 119874 (13)

As 1198692(119869119896

2119872) = 0 by Proposition 15 there exists a nonzero

natural number 1198991015840 such that 1198691198962119872 ≃ oplus

1198991015840

119894=1(1205901minus 1)119896119894119870[1198661]

0 le 119896119894lt 119901119899 as 119870[119866]-modules where the action of 119866

2on

oplus1198991015840

119894=1(1205901minus 1)119896119894119870[1198661] is trivial 119872119869

119896

2119872 is a 119870[119866]-module of

type 119896 with 1198691198962(119872119869119896

2119872) = 0 more119872119869

119896

2119872darr1198661

is 119877-free and119896 = 119901

119903 with 0 lt 119903 le 119899 Then Theorem 9 shows that119872119869119896

2119872

is a free119870[1198661times1198672]-moduleTherefore there exists a nonzero

natural number 11989910158401015840 such that119872119869119896

2119872 ≃ (119870[119866

1times1198672])11989910158401015840

Therest is a simple application of Proposition 12

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S O Kaptanoglu ldquoStructure and detection theorems fork[C2timesC4]-modulesrdquo The Rendiconti del Seminario Matematico

della Universita di Padova vol 123 pp 169ndash189 2010[2] P Webb Finite Group Representations for the Pure Mathemati-

cian University of Minnesota 2013[3] J Thevenaz Representations lineaires de groupes finis en car-

acteristique p119903 Universite de Geneve 1980[4] K W Gruenberg Cohomological Topics in Group Theory vol

143 of Lecture Notes in Mathematics Springer Berlin Germany1970

[5] G Karpilovsky The Jacobson Radical of Group Algebras NotasDeMathematica vol 135 Elsevier Science NewYork NY USA1987

[6] J Thevenaz ldquoRepresentations of finite groups in characteristic119901119903rdquo Journal of Algebra vol 72 no 2 pp 478ndash500 1981

[7] C W Curtis and I Reiner Methods of Representation TheoryWiley-Interscience New York NY USA 1981

[8] M E Charkani and S Bouhamidi ldquoModular representationsof Loewy length twordquo International Journal of Mathematics andMathematical Sciences no 70 pp 4399ndash4408 2003

[9] B Huppert and N Blackburn Finite Groups II Springer BerlinGermany 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Algebra 5

Theorem 16 Let 119872 be a finitely generated 119870[119866]-module If1198692

1119872 = 0 then there exist two nonzero natural numbers 1198991015840 and

11989910158401015840 and two119870[119866]-modules119872

1= oplus1198991015840

119894=1(1205902minus1)119896119894119870[1198662] 0 le 119896

119894lt

119901119899 and 119872

2= oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] 0 le 119896

1015840

119894lt 119901119899 where the

action of 1198661on1198721and119872

2is trivial and there is a derivation

119865 from 119870[119866] in119867119900119898119870(11987221198721) such that119872 cong 119872

1times1198651198722

Proof We have the exact sequence of 119870[119866]-modules 119874 rarr

1198691119872 997893rarr 119872 rarr 119872119869

1119872 rarr 119874 As 1198692

1119872 = 0 119869

1(1198691119872) = 0

So by Proposition 15 there exists a nonzero natural number1198991015840 such that 119869

1119872 cong oplus

1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 119901119899 as

119870[119866]-modules where the action of1198661on oplus119899

1015840

119894=1(1205902minus1)119896119894119870[1198662]

is trivial We set 1198721

= oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] We have

1198691(1198721198691119872) = 0 So by Proposition 15 there exists a nonzero

natural number 11989910158401015840 such that1198721198691119872 cong oplus

11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662]

0 le 1198961015840

119894lt 119901119899 as 119870[119866]-modules where the action of 119866

1

on oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] is trivial We set 119872

2= oplus11989910158401015840

119894=1(1205902minus

1)1198961015840

119894119870[1198662] Then Proposition 12 shows that 119872 cong 119872

1times1198651198722

for a derivation 119865 from119870[119866] in Hom119870(11987221198721)

If 119901 = 2 1198661= 1198622 and 119866

2= 1198622119899 then we have the

following corollary

Corollary 17 For all finitely generated 119870[119866]-modules thereexist two nonzero natural numbers 1198991015840 and 11989910158401015840 and two 119870[119866]-modules 119872

1= oplus1198991015840

119894=1(1205902minus 1)119896119894119870[1198662] 0 le 119896

119894lt 2119899 and

1198722= oplus11989910158401015840

119894=1(1205902minus 1)1198961015840

119894119870[1198662] 0 le 119896

1015840

119894lt 2119899 where the action

of1198661on1198721and119872

2is trivial and there is a derivation 119865 from

119870[119866] in119867119900119898119870(11987221198721) such that119872 cong 119872

1times1198651198722

Proof We have 1198691= (1205901minus 1)119870[119866

1] and as (120590

1minus 1)2= 0 since

the field 119870 is of characteristic 119901 = 2 11986921= 0 So 1198692

1119872 = 0 for

119870[119866]-module of finite type119872The rest is a simple applicationof Theorem 16

Now we return to cases 1198661= 119862119901119898 and 119866

2= 119862119901119899 where

1198662is generated by an element 120590

2 and let 119869

2be the Jacobson

radical of 119870[1198662] For an integer 119896 = 119901

119903 with 0 lt 119903 le 119899 andfor the subgroup 119867

2of 1198662generated by 120590119901

119899minus119903

2 we have the

following result

Theorem 18 Let119872 be a finitely generated119870[119866]-module with119869119896+1

2119872 = 0 If 119872119869

119896

2119872darr1198661

is 119877-free and of type 119896 then thereexist two nonzero natural numbers 1198991015840 and 11989910158401015840 and two 119870[119866]-modules119872

1= oplus1198991015840

119894=1(1205901minus 1)119896119894119870[1198661] 0 le 119896

119894lt 119901119899 and119872

2=

(119870[1198661times 1198672])11989910158401015840

where the action of 1198662on1198721is trivial and

there is a derivation 119865 from119870[119866] in119867119900119898119870(11987221198721) such that

119872 cong 1198721times1198651198722

Proof We have the following exact sequence

119874 997888rarr 119869119896

2119872 997893rarr 119872 997888rarr

119872

119869119896

2119872

997888rarr 119874 (13)

As 1198692(119869119896

2119872) = 0 by Proposition 15 there exists a nonzero

natural number 1198991015840 such that 1198691198962119872 ≃ oplus

1198991015840

119894=1(1205901minus 1)119896119894119870[1198661]

0 le 119896119894lt 119901119899 as 119870[119866]-modules where the action of 119866

2on

oplus1198991015840

119894=1(1205901minus 1)119896119894119870[1198661] is trivial 119872119869

119896

2119872 is a 119870[119866]-module of

type 119896 with 1198691198962(119872119869119896

2119872) = 0 more119872119869

119896

2119872darr1198661

is 119877-free and119896 = 119901

119903 with 0 lt 119903 le 119899 Then Theorem 9 shows that119872119869119896

2119872

is a free119870[1198661times1198672]-moduleTherefore there exists a nonzero

natural number 11989910158401015840 such that119872119869119896

2119872 ≃ (119870[119866

1times1198672])11989910158401015840

Therest is a simple application of Proposition 12

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S O Kaptanoglu ldquoStructure and detection theorems fork[C2timesC4]-modulesrdquo The Rendiconti del Seminario Matematico

della Universita di Padova vol 123 pp 169ndash189 2010[2] P Webb Finite Group Representations for the Pure Mathemati-

cian University of Minnesota 2013[3] J Thevenaz Representations lineaires de groupes finis en car-

acteristique p119903 Universite de Geneve 1980[4] K W Gruenberg Cohomological Topics in Group Theory vol

143 of Lecture Notes in Mathematics Springer Berlin Germany1970

[5] G Karpilovsky The Jacobson Radical of Group Algebras NotasDeMathematica vol 135 Elsevier Science NewYork NY USA1987

[6] J Thevenaz ldquoRepresentations of finite groups in characteristic119901119903rdquo Journal of Algebra vol 72 no 2 pp 478ndash500 1981

[7] C W Curtis and I Reiner Methods of Representation TheoryWiley-Interscience New York NY USA 1981

[8] M E Charkani and S Bouhamidi ldquoModular representationsof Loewy length twordquo International Journal of Mathematics andMathematical Sciences no 70 pp 4399ndash4408 2003

[9] B Huppert and N Blackburn Finite Groups II Springer BerlinGermany 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of