P E T E R S C H M I D

F R A T T I N I A N p - G R O U P S

To Helmut Salzmann on his 60th birthday

ABSTRACT. The notion of a Frattinian p-group generalizes that of an extra-special p-group. We prove a central decomposition theorem and describe some relations to Frattini extensions and to automorphisms of finite p-groups.

1. I N T R O D U C T I O N

There exists no systematic theory of finite p-groups. Most of the deeper investigations are concerned with special classes of p-groups. In this note we suggest the notion of a Frattinian p-group which is stimulated from properties of extra-special p-groups. It leads us to an interesting decom- position which may be a guide for further studies.

Fix a finite p-group G for some prime p. We call G Frattinian provided its centre Z(G) ~ Z(M) for all maximal subgroups M of G. This is some kind of non-degeneracy condition (for non-abelian p-groups). Extra-special p-groups are indeed Frattinian.

THEOREM. Suppose G is a non-abelian Frattinian p-group. Then one of the following holds:

(i) G is the central product of non-abelian p-groups of order pzIZ(G)[, amalgamating their centres.

(ii) G = E. F is the central product of Frattinian subgroups E, F where Cv(Z(@(F)) ) = ~(F) and where E = CG(F ) satisfies ~(E) ~ Z(G).

Here ~(F) denotes the Frattini subgroup of F, Observe that the two statements do exclude each other. In (ii) either E = Z(G) or E is a central product as in (i). In particular, IE/Z(G)I is a square which turns out to be an invariant for G.

A Frattinian p-group G satisfying CG(Z(O(G)))= @(G) will be called strongly Frattinian, for convenience. The author proved in [7] that H~(G/@(G), Z(@(G))) ¢ 0 if G is a non-abelian regular p-group. Whenever this cohomology group vanishes, either G is elementary or strongly Frattinian (see Proposition 2). This was another motivation for introducing the concept.

Of course, the variety of Frattinian p-groups is not closed with respect to the usual group theoretic closure operations. Also, some important types of

Geometriae Dedicata 36: 359-364, 1990. © 1990 Kluwer Academic Publishers. Printed in the Netherlands.

360 PETER SCHMID

p-groups, e.g. those of maximal class (and order >~ p4), are not Frattinian. On

the other hand, we show that every non-cyclic p-group G is an epimorphic image of a strongly Frattinian one having the same minimum number of

generators. In fact, the maximal Frattini p-extension of G turns out to be

strongly Frattinian (see Proposition 1). We finally give an application concerning p-automorphisms. The theorem

enables one to give a very lucid approach to the celebrated result by Gaschfitz [3] and its various improvements.

2. SOME CENTRAL DECOMPOSITIONS

Let M be a maximal subgroup of the finite p-group G. Then either its

centralizer CG(M ) = Z(M) or CG(M ) = Z(G) is a supplement to M in G. Also,

if Z(M) ~ Z(G), then M = Ca(Z(M)). These are the basic ingredients in the

P R O O F OF THE THEOR EM. Let G be non-abelian and Frattinian.

Suppose first that Z(M) ~ Z(G).~(G) for any maximal subgroup M ~_ Z(G) of G. Then M = CG(Z(M)) for all these M, and the join J of all their centres Z(M) is a subgroup of Z(G).Z(@(G)) satisfying

CG(J) ~- ~ C6(Z(M)) = Z(G).@(G). M

Thus CG(Z(@(G))) = Z(G)'@(G) in this case. Let F be a minimal supplement to Z(G) in G. Then F is normal in G and Z(F) = Z(G) n F is contained in O(F),

Moreover, CG(F) = Z(G) and Cv(Z(O(F))) = O(F). If X is a maximal sub-

group of F, then M = Z(G). X is a maximal subgroup of G. We conclude that F is strongly Frattinian.

So we may assume that there is a maximal subgroup M ~_ Z(G) of G such

that Z(M) ~ Z(G).~(G). Then we find a maximal subgroup N ~ Z(G) of G with N ~ Z(M). We know that M = Ca(Z(M)) and N = Ca(Z(N)). We assert that Z(M) ~ N = Z(G). Otherwise we would have

and

M = CG(Z(M) ¢~ N) ~_ Co(N) = Z(N)

N -- C . ( Z ( N ) ) =_ CdM) = Z(M),

a contradiction. Similarly we have G = Z(N). M and Z(N) a M = Z(G). We

infer that E1 = Z(M).Z(N) is a non-abelian normal subgroup of G with E~/Z(G) being elementary of order p2. Further, Z(E~) = Z(G) and

G~ = Co(E1) = MeaN

F R A T T I N I A N p - G R O U P S 361

is a supplement to E 1 in G. Hence Z(G1) = Z(G) also. If M 1 ~_ Z(G) is a

maximal subgroup of GI, then El"M1 is a maximal subgroup of G and therefore

Z(MI) = C~I(MI) = Ca(E 1 • M,) = Z(E~ "M1)

properly contains Z(G). Thus G 1 is Frattinian. We may proceed as before. If there is a maximal subgroup MI ~ Z(G) of

G 1 with Z(M~)~Z(GO'O(GO, we obtain a central decomposition G~ = E a ' G 2 where Z ( E 2 ) = Z(G)= Z(G2) and IEzl = p2]Z(G)[, and so on. This process must stop after finitely many steps, say n. Then we have a central decomposition G = E" G, where E = Ez -.. E, is itself a central product over Z(G) = Z(E) and G, = CG(E) does not have such a maximal subgroup. Either G, = Z(G) or any minimal supplement F to Z(G) = Z(G.) in G, is strongly Frattinian, as above.

It is clear that also E is Frattinian (putting E = Z(G) in the first case). Of course, the central decomposition G = E . F will not be unique in general.

Since Z(G).~(F) = Z(G).~b(G), the centralizer CG(cb(F)) = C~(q~(G)) certainly is independent of the choice of F. But also

CG(¢(F)) = E. Z(~P(F)) ~_ E. dP(G).

We conclude that, at least, the order of E is an invariant of G. []

3. F R A T T I N I EXTENSIONS

Let G be a finite non-cyclic p-group, and let A = Ap(G) be its Frattini module over ~:p. (This A is the second Heller module of the trivial ~:pG-module; see [-4] for a detailed discussion.) By a result of Gaschfitz [-2] there is, up to

isomorphism over G, a unique group extension A~--~G--~ G such that A _~ @(G), and every Frattini extension of G with elementary abelian kernel (lying in the Frattini subgroup of the extension group) is an epimorphic image of G (cf. also [5, p. 272]).

Of course, this holds true for all finite groups. In our situation, if

R ~ F --~ G is a minimal free presentation of G, we have A ~- R/R'R p and G "~ F/R'R p. So from Schreier's subgroup theorem it follows that

dim Ap(G) = 1 + ]GI(d(G) - 1),

where d(G) is the minimum number of generators for G (and (~).

P R OP OS ITI ON 1. (~ is strongly Frattinian. More precisely, for every subgroup I1 ~_ A of G we have Z(I~) ~_ A and C~(Z(I~)) = lq.

Proof As G is assumed non-cyclic, A = Ap(G) is a faithful FpG-module by

362 PETER SCHMID

[4, Th. 3]. Thus C~(A) = A and Z(/ t) c A is the fixed module A H = CA(H) of

A under H = It/A. We claim that

dim A H = 1 + [G:Ht(d(G)- 1).

Having established this, the result readily follows. For /~ = C~(A n) obviously contains /~ and A ~ Z ( K ) . But K = K / A contains H and so Z(/~) = A K _ An. Hence A n = A K, and the dimension formula (for H and K)

yields that H = K. In order to compute dim A n we make use of some further results from [4].

First, as an ~pH-module, A = B • Q where B ~_ Ap(H) and Q is a free ~:pH-

module. Also, B n is the socle of B and therefore

dim B n = dim Hi(H, B) = d(H).

By the very structure of the regular module ~:pH, dim(0ZpH)n = 1 and so

dim Q = [H[' dim QH. On the other hand, we know that

dim Q = dim Ap(G) - dim Ap(H)

--- IHI(IG :HI (d (G) - 1) - d ( H ) + 1).

We conclude that dim A n = dim B/~ + dim Qn is as claimed. []

The dimension of the Frattini module A = Ap(G) increases rapidly with IGI

and d(G). As shown in [4], there are faithful quotient modules B of A of fairly

small dimensions. For example, if G is elementary abelian of order p2 with p

odd, the least possible such dimension is dim B = 3. There is actually a

strongly Frattinian p-group, B ~ P --~ G of order IPI = pS, which is as small as possible.

REMARK. The formula for dim A n enables us to compute 1-cohomology. Suppose H = It/A is a proper normal subgroup of G. Then Hi(G/H, A H) ~ 0 for otherwise A n = Z(/t) is a free g:p[G/H]-module and so dim A H is divisible

by IG:HI. Application of a result by Tate ([5, p. 255]), and of the inflation- restriction sequence, shows that dim H~(G/H,A H) ~< 1. Thus we have equality.

4. C O H O M O L O G Y AND A U T O M O R P H I S M S

If M is a normal subgroup of some group G, then HI(G/M, Z(M)) is isomorphic to the group of all automorphisms of G centralizing M and G/M (even G/Z(M)) modulo the group of all inner automorphisms induced by

Z(M) (e.g. see [5, p. 46]). Thus if C~(M) = Z(M), the cohomology group is isomorphic to a subgroup of Aut(G)/Inn(G).

FRATTINIAN p-GROUPS 363

P R O P O S I T I O N 2. Let G be a finite p-group. I f HI(G/@(G), Z(@(G))) = O, then either G is elementary abelian or strongly Frattinian.

Proof Assume ~ ( G ) # 1. By Gaschfi tz 's result [3], Z(dO(G)) is a cohomological ly trivial G/~(G)-module. This implies that G/~(G) acts faith-

fully on Z(@(G)) and that CG(Z(M)) = M for all subgroups M ___ q)(G) of G (cf. [7, Prop. 1]). Hence G is s t rongly Frat t inian. [ ]

It has been shown in [1] that there are (non-regular) p-groups where this

cohomology does vanish.

P R O P O S I T I O N 3. I f G is a strongly Frattinian p-oroup, the normal p- subgroup of Aut(G) consistin9 of all automorphisms centralizin9 G/@(G) and Z(@(G)) is not inner.

Proof Let M be a maximal abelian normal subgroup of G containing

Z(@(G))). Then

M = Ca(M) ~_ C~(Z(O(G))) = O(G).

It follows that the group extension M~-* G--* G/M does not split. Thus H2(G/M, M)v~ 0 and therefore HI(G/M, M ) ~ 0 by [3]. This yields the

statement. [ ]

If the non-trivial p-group G possesses a maximal subgroup M for which

Z(M) = Z(G), then even Hom(G/M, Z(G) n (I)(G)) ~ 0, which cor responds to the au tomorph i sms of G centralizing M and G/Z(G) n (I)(G). Also, if G = E- F

is a central product , every pair of au tomorph i sms of E and F which agree on E n F extends uniquely to an a u t o m o r p h i s m of G. Thus in view of the

theorem, Propos i t ion 3 gives at once Gaschfi tz 's result on outer a u t o m o r p h -

isms of p-groups as well as the s trengthening obta ined in [6].

In part icular, Proposi t ion 3 holds for all p-groups G excluding the cases where @(G) = Z(G) is cyclic or (I)(G) = 1.

REFERENCES

1. Caranti, A., 'On the automorphism group of certain p-groups of class 4', Boll. Un. Mat. ltal. B (6) (1983), 605-615.

2. Gaschfitz, W., 'Uber modulare Darstellungen endlicher Gruppen, die von freien Gruppen induziert werden', Math. Z. 60 (1954), 274-286.

3. GaschOtz, W., 'Kohomologische Trivialit/iten und/iu[}ere Automorphismen von p-Gruppen', Math. Z. 88 (1965), 432-433.

4. Griess, R. L. and Schmid, P., 'The Frattini module', Arch. Math. 30 (1978), 256-266. 5. Gruenberg, K. W., 'Cohomological topics in group theory', Springer Lecture Notes in Math.

143 (1970). 6. MOiler, O., 'On p-automorphisms of finite p-groups', Arch. Math. 32 (1979), 533-538. 7. Schmid, P., 'A cohomological property of regular p-groups', Math. Z. 175 (1980), 1-3.

364 PETER SCHMID

(Received, September 15, 1989)

Author's address:

Peter Schmid, Mathematisches Institut der

Universit/~t Tfibingen, Auf der Morgenstelle 10, D-7400 Tiibingen, F.R.G.