2. S. Grosser and M. Moskowitz, "Compactness conditions in topological groups," J. Reine Angew. Math., 246, 1-40 (1971).

3. O.V. Mel'nikov, "Groups of automorphisms of compact, totally disconnected groups," Dokl. Akad. Nauk BSSR, 16, No. 9, 777-780 (1972).

4. C. Wells, "Automorphisms of group extensions," Trans. Amer. Math. Soc., 155, 189-194 (1971).

5. S. Lang, Report sur la Cohomologie des Groupes, Benjamin, New York--Amsterdam (1966). 6. L. Fuchs, Abelian Groups, Publ. Hause Hung. Acad. Sci., Budapest (1958). 7. S. MacLane, Homology, Springer, Berlin (1963).

THE ~ "LENGTH OF FINITE GROUPS

V. I. Kharlamova UDC 519.4

L. A. Shemetkov [i] defined the concept of the p-length of an arbitrary finite group, generalizing the concept of the p-length of a p-solvable finite group of Hall--Higman [2]. In this article we introduce a more general concept of the ~ - length of an arbitrary finite group with respect to a certain formation ~ , and bounds on ~ -lengths and p-lengths are given for arbitrary finite groups.

The definitions and notations used in the present work may be found in the review [3]. However, we need a series of additional definitions and notations.

A class of groups ~ is called a homomorph if it is closed with respect to the taking of epimorphic images. A class of groups ~ is called a formation if ~ is a homomorph and if from G/N~ and G/B~ it always follows that G/AD B ~ [4, p. 696, Definition 7.1].

If ~ contains all the invariant subgroups of its groups then the formation ~ is called invariantly closed [5]. The formation ~ is called saturated if from G/O (G)~ ~ it follows that G E~ [4, p. 696, Definition 7.2].

Let the function f associate with every prime number p a formation f(p), which might be empty. An invariant subgroup K of the group G is called ~ -imbedded in G [4, p. 698] if for any factor H/L of a chief series for G such that If~K the following condition is re- alized: G/C~(H/L)~](p) for every prime divisor p of the order of H/L. The class ~ of all groups 9 -imbedded in themselves constitutes a formation which is called a local formation or, more to the point, the formation locally defined by the formations f(p) [4, p. 696, Defi- nition 7.3].

G~ denotes the ~-co-radical of the group G, i.e., the smallest invariant of G such

that G/G~E ~ [5].

Definition l. Let

G = G o ~ 6 1 ~ . . . ~ G t =E , t > 0 (1)

be a certain chief series of the group G. The product of all I G~: Gi+1[ such that Gi/G~+I ~ ~, where ~ is a certan formation, we call the degree of ~ -saturation of the group G and de- note it by s~(G) . We will define s~(G) -- i if none of the factors of the series (i) be- longs to ~ .

G0mel Branch of the Mathematicslnstitute, Academy of Sciences of the BSSR. Trans- lated from Matematicheskii Zametki, Vol. 19, No. 5, pp. 745-754, May, 1976. Original arti- cle submitted July Ii, 1975.

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mectmnical, photocopying, microfilming, recording or otherwise, wi thout written permission o f the publisher_ A copy o f this article is available f rom the publisher for $ 7. 50.

44-~

We note that if the ~-formation is the solvable q-groups, then the degree of ~-satura- tion is called degree of ~-solvability of the group G (see [i]).

LEMMA i. Let the group G = A-B and A be an invariant subgroup of G. Then s~(G) = s~ (A).s~ (B)

s~ (A N B)

P r o o f . I t i s o b v i o u s t h a t si~(G ) ~ s~(G/A) .s~ (A) . But s v ( G /A ) = s v ( A B / A ) = s v ( B / A N B ) = s,~ (B): s ~ ( A ( ] B ) �9 Thus , sv(G ) ~ s ~ ( A ) . s ~ ( B ) : s i~(A~]B ) . The temma i s p r o v e d .

From Lemma 1 follows

LEMMA 2. The product of all the invariant subgroups of the group G, having degree of -saturation equal to i, is a characteristic subgroup, the degree of ~-saturation of which

is also equal to i.

We denote by H~ (G) the product of all invariant subgroups K of the group G which satisfy the following condition: ]K/K~ [= s~ (K).

LEMMA 3. If ~ is an invariant by closed formation, then ]H~ (G)/(f-f~ (G))~[- s~ (/f~ (G)).

Proof. Let A and B be invariant subgroups of the group G such that ]N/A~ [ = s~ (A) and [B/B~ [ =s~(B). By Lemma 1 the degree of ~ -saturation of the subgroup r ~ is equal to i. In addition, Ar 9 andBC/C~ 9; thus, AB/C~ 9. The lemma is proved.

From Lemma 3 it follows that H~ (G) is a characteristic subgroup of the group G, and (H~(G))~ is the product of all invariant subgroups of the group G which have degree of ~ - saturation equal to i.

The factor A/B we will call ~-saturated if s~ (A/B) = ]A/B [.

Definition 2. We define the upper ~-series of the group G to be the series

E = A o ~ B o C A I C B I ~ A ~ . . . c A l c - ~ B t - - - - G , (* )

in which Bi/A i is the product of all the invariant subgroups of G/Ai having degree of 9- saturation equaling 1 and in which Ai+~/Bi is the product of all invariant subgroups K/B i of G/B i such that s~ (K/Bi) := [K/Bi [, and where ~ is the smallest whole number for which B~ = G.

Definition 3. The number of nonidentity 9-saturated factors of the series (*) where is an invariantly closed formation is called the ~-length of the group G and is denoted

by l~ (G) . Obviously, l~ (G) ~ I.

We note that if ~ is the formation of p-groups and G is a p-solvable finite group then from the definition of the ~-length is obtained the definition of the p-length of p-solvable group of Hall and Higman [4, p. 688, Definition 6.1]. If ~ is the formation of p-groups and G is an arbitrary finite group then from Definition 3 we obtain the definition of p-lengths of arbitrary finite groups of L. A. Shemetkov [I].

For the designation of a local formation the notation ~ ~-(f (p)> is used. In particular ----</(p) ~= ~> is a formation locally defined by a nonempty formation for every p.

Following M. Hale [6] and Shemetkov [5], we call the maximal subgroup M of the group G -normal if G/cor~]l~ ~,where corGM is the core of the subgroup M in G, i.e., the intersec-

tion of all subgroups conjugate with M in G. A maximal subgroup which is not ~-normal is called ~, -abnormal.

This definition in the class of solvable groups coincides with the definition of ~- normality in [7].

We define on the finite groups the whole-number function ~ (G) in the following way.

Definition 4~ ~ (G) ~ ~l if any maximal chain of the group G of length n contains a proper subgroup ~-normal in the preceding subgroup and if there exists at least one maxi- mal chain of length n -- 1 in which each subgroup is ~ -abnormal in its preceding subgroup.

From this definition it follows that if ~ (G)= n,then in the group G there is always at least one m~ximal chain of length n in which only the last group is ~-normal in the pre- ceding.

443

LEMMA 4. If H is not an ~-normal maximal subgroup of the group G then %~ (H)~ ~,~ (G)

Proof. Let ~ (G) ----n. We consider an arbitrary maximal chain of the group G of length n passing through the subgroup H. From Definition 4 it follows that it contains a subgroup R which is ~ -normal in the subgroup preceding it. Hence, an arbitrary maximal chain of length n -- 1 of the subgroup H contains a subgroup ~-normal in the preceding subgroup, i.e., ~(H)~ n-- I = ~(~ -- I . The lemma is proved.

LEMMA 5. If K is an invariant subgroup of the group G then ~ (G/K)~ ~ (G).

Proof. We assume that re = ~ (G/K)~ ~ (G)= n.. We consider the maximal chain of the

factor-group G/K of length m:

(a ) G / K = Go /K ~ G J K ~ . . . ~ GIn~K,

in which only the subgroup Gm/K is ~-normal in Gm_~/K. From chain (a) we may obtain the maximal chain of the group G:

(b ) G = G o ~ G 1 ~ �9 �9 �9 ~ Gn . .~ . �9 �9 ~ G m.

By assumption in chain (b) there is a subgroup G~(I~ i~ n), ~ -normal in Gi-x , i.e., G~_~/ cor~ (G~) ~ ~, Then

G i _ ~ / e o r o (Gi) ------- G~_JK/corc, (G~)/K ~ ~ .

Hence, the subgroup G~/K where i~n, ~ -normal in Gi-~/K, is a contradiction. The lemma is

proved.

We recall that A~ (G) denotes the intersection of all ~ -abnormal maximal subgroups of

of the group G.

The following lemma follows from Theorem 1 of [8] by M. V. Sel'kin.

LEMMA 6. Let ~ ~- ~ ~)-5~0> be a certain local and invariantly closed formation.

Then ~(G)~ ~ .

THEOREM i. Let ~ = </~)~> be a local and invariantly closed formation. For an

arbitrary finite group G we have I~(G)~(G).

Proof. Let ~(G) = I; then all the maximal subgroups of the group G are ~ -normal. Thus, for any maximal subgroup M of G we have G/co~M~ ~ . Consequently, G/~ (G)~ , from which we find that G~. Hence, if ~(G) = i then I~(G) = i.

We assume that the theorem is true for all groups K such that ~(K) = n--1 and is not true for some groups L for which T~ (L) = n . Among such groups we choose the group G of the

smallest order.

We consider the upper ~-series of the group G:

E = A o ~ _ _ B o C A 1 c B I ~ A ~ _ ~ _ . . . c A t ~ _ B I = G.

If the subgroup Bo, which is a maximal invariant subgroup of the group G with degree of ~ - saturation equal to I, is different from E, then ~(G) -- l~(G/Bo) ~ T~(G/B0)~T~(G). In this case the theorem is correct. Now let Bo = E. Then A1~Ewill be a maximal invariant sub- group of G such that ~ (At) = IAII. The subgroup B:/A: is invariant in G/A~ and has degree of

-saturation equal to i. If B~Au(G), then by Lemma 6 we would find that BI~A~(G)~G. Then BI/AI~, which would be a contradiction. This means that G = B~M where M is some

-abnormal maximal subgroup of G. We have

By assumption (G) - - t = l~ (G/BO < ~ (G/B,) < ~ (G) - - t .

From this it follows that ~ ( G ) . ~ ( G ) . The theorem is proved.

Let OP(G) denote the intersection of all those invariant subgroups of the group G the factor-groups by which are p'-groups and let Fo(G) denote the product of all those invariant subgroups H of the group G which possess the ~ollowing property: IH/OP(H)] = sp(H). From Lemma 3 it follows that Fp(G) possesses this property and is, consequently, a radical.

4 4 4

Definition 5. We will say that ff1~ (G) : n if each maximal chain of pd-subgroups of the group G of length n contains a proper subgroup, invariant in the subgroup preceding it and if there exists at least one maximal chain of pd-subgroups of length n -- 1 in which all sub- groups are noninvariant in the preceding subgroups. We define ~p (G) = 0 if G is a p'-group.

THEOREM 2. For an arbitrary finite group G we have l~ (G)~ ~p (G).

Proof. If ~p (G) ---- i we have G v <3 G, and the theorem is obvious. We assume that the theorem is true for all groups K in which ~p (K)~ n- I and not true for some groups L which have ~v (L)~--n. Among such groups we choose the group G of the lowest order.

If G is not a pd-group then the correctness of the theorem follows from the definition of lv(G): in this case lp(G) = 0. Therefore, we further assume that p divides ]G I. We consider the series of subgroups

E -= F o ~ F 1 C �9 �9 �9 ~ El ~ G,

where F~+I/F ~ = Fv (GIFt), and 1 = lp(G). Then FI = Fp(G). From Lemma 2 it follows that OP(FI) is the product of all those invariant subgroups of the group G which have measure of p- solvability equal to i. We assume that 0 v (FI) =/= E. Then Ip (G) ~ Ip (G/O p (F1)) ~ ~Pv ( G/Ov (fl)) ~p(G) , i.e., in this case the theorem is correct. Now let OP(Fx) = E. Then the subgroup F~----Fv(G) DE is a p-subgroup and fD(G)~F I. By definition Fi/F~ = Fp(G/GI). The subgroup OP(Fp(G/FI)) = N/Fx is the product of all those invariant subgroups of the group G/F~ which have degrees of p-solvability equal to i. If the subgroup N were nilpotent then in G there would be an invariant p'-subgroupwhich would contradict the assumption. Therefore, N is not nilpotent and this means that N is not contained in A(G). Then there is in the group G an invariant maximal subgroup M such that G = MN. If all the maximal subgroups of G not con- taining N are p' -groups then Gp c N, and we easily find that Gp <3 G, i.e., lp(G) = i. Thus, let M be a pd-group. This means that

c~ v ( G / N ) = (Pv (Il l~N(-] 31) ~ %, (M) < q~v(C)--t.

By induction we find that

l v (G) - - i = 1 v ( G / N ) ~.~ fr ( G / N ) < q)p (G) - - i .

From this it follows that Ip (G) ~ q)p (G). The theorem is proved.

COROLLARY I. Let every maximal chain of pd-subgroups of the group G of length n con- tain a proper nilpotent subgroup. Then l~ (G) ~ n ~- I.

THEOREM 3. If every p-subgroup of the group G is Sylow in a certain subnormal subgroup of G then Ip(G) = i.

Proof. Among the groups for which the theorem is not true we choose the group G of the smallest order.

If the subgroup N is invariant in G then it is obvious that the hypothesis of the theorem holds for the factor-group G/N. Let the subgroup K be the product of all the invar- iant subgroups of the group G having degree of p-solvability i. If K ~= E then lp(G/K) = Ip(G) = i. Therefore, we further assume that K = E. Then G has an invariant p-subgroup different from E. We denote by P the greatest invariant p-subgroup of the group G. We as- sume that P ~ qb (G). Hence, in G there is a maximal subgroup M such that G = PM. If M is a p'-subgroup then P will be a Sylow p-subgroup of G, in which case the theorem is obvious. Thus, let ~ [ p :~ E be a Sylow p-subgroup of M. By the hypothesis, Mp is Sylow in a certain subnormal subgroup L of G. It is obvious that L is a proper subgroup of the group G. But then in G there exists a proper invariant subgroup K such that 3[p ~ L ~ K C G. Every p- subgroup of K will be Sylow in a certain subnormal subgroup R of G. But then it will be Sylow also in the subgroup K ~ R which is subnormal in K. Thus, by assumption lp(K) = i.

We assume that PK = G. In this case we have

lp (G) - - I = bp ( G / K ) = I v ( K / K ~ P ) .

It is clear that K does not have invariant subgroups with degree of p-solvability equal to i, and the /f ~ P is the largest invariant p-subgroup of K, i.e., Ip (K) =l v(K/K~ I ) )~I. But lp(K) = i, from which it follows that lp(G) = i. If PK =j= G then PK is a proper invariant subgroup of G containing Gp. By assumption lp(PK) = i, but lp(PK) = lp(G); i.e., in this case the theorem is true.

445

Now we assume that P ~ ~)(G). We denote by N/P the product of all those invariant sub- groups of the factor-group G/P the degree of p-solvability of which is equal to I. Then IN Ip< IG I p; in the contrary case sp(G/P) = i, i.e., Ip(G) = I. In addition, IP I< IN Ip, otherwise by the well-known Lemma 3.2.1 of S. A. Chunikhin of [9] we would find that NG(H) ~ (PH) = G where H is a Hall p-complement of N. From this it would follow that NG(H)-~(G) = G, i.e., H would be an invariant p-subgroup of G, a contradiction.

We assume that N ~__ fp (~ where Fp(G) is the intersection of all maximal subgroups of the group G the indices of which are not divisible by the prime number p. By the theorem of W. Deskins in [i0] the subgroup ~n (G): and hence also N are solvable and moreover Np <:] G a contradiction. Thus, G = N.M where IG:M[ is divisible by p. By the hypothesis Mp is Sylow in a certain subnormal subgroup L of G. It is obvious that L is a proper subgroup of the group G. Thus, in G there exists a proper invariant subgroup K such that Mp ~ L ~ K C G. Since IK I < ~G I, it follows that ~p(K) = i. From this it follows that the degree of p- solvability of the subgroup K/P is ~qual to i, but then K c: N. This means that Mp = Np and Np = Gp, which is again a contradiction. The theorem is proved.

We note that the converse theorem for arbitrary finite groups is not true; for a counter-example it is possible to use the group PSL (2, 5). Under the further hypothesis of p-solvability of the group G this theorem is proved in [Ii] by G. A. Chambers.

In the following theorem we will say that Ip(G) = i if and only if GpK <] G, where K is the product of all the invariant subgroups of G with a degree of p-solvability of i.

In the work of R. Schmidt [12] the definition of a modular subgroup is given.

THEOREM 4. If every p-subgroup of the group G, where p is the largest prime number dividing the order G, is Sylow in a certain modular subgroup of G then Ip(G) = I.

Proof. Let G be a group of smallest order for which the theorem is not true.

We assume that K is the largest invariant subgroup of G with degree of p-solvability I. We consider the factor-group G/K. If H/K is an arbitrary p-subgroup of G/K then in G there is a p-subgroup P such that PK = H and P is a Sylow p-subgroup of the modular subgroup M of G. Then H/K will be Sylow in the modular subgroup MK/K of G/K. By assumption Ip(G) = i. Therefore K = E.

consequently, in G there exists a largest invariant p-subgroup J different from E. We assume that ] ~ (D (G). Then G = JM where M is some maximal subgroup of G. It is obvi- ous that Mp ~= E. By the hypothesis Mp is Sylow in the modular subgroup L of the group G. Let L ~ H where H is some modular maximal subgroup of G. If /f <~ G, then obviously in H there are noninvariant subgroups having degree of p-solvability equal to i. Let P be an arbitrary p-subgroup of H which is Sylow in the modular subgroup F of G. Then P will be a Sylow p-subgroup also in the modular subgroup H ~ F of H. By assumption Ip(H) - i, i.e., Hp <::] H and Hp <::] G. But since JMp = Gp and tVf p ~ H_p, it follows that .]Hp = Gp <::] G. Thus, in this case the theorem is true. Now let H be noninvariant in G. Then by Lemma 1 of [12] we find that IG:corGH I -- pq where p and q are distinct prime numbers. If ]H:CorGH[ = q then in G there exists an invariant subgroup R of index q; see Corollary (3) of [I0]. As q=/=p we haveGp~R. By assumption G~<:~R and hence, Gp<:]G. Consequently, [H:CorGH I = p. Since p is the largest prime number dividing [G I and the factor group G/corGH is non-Abel- ian of order pq then it is clear that H<~ G. Again there is a contradiction.

Now we assume that J ~ dO (G). We consider G/J. Let H/J be a proper invariant subgroup of G/J. Then H is invariant in G. Since the hypothesis of the theorem is satisfied for H, it follows that H p < 3 G a n d H p ~ J . Consequently, H----Rp>~R, where IT is a p'-group. But then it is easy to show that H <~G. This gives a contradiction with the above. Hence, the factor-group G/J is simple. Let P be a proper p-subgroup of Gp not belonging to J. By the hypothesis P will be a Sylow subgroup in the modular subgroup K of G. If K <~ G then by the assumption we have _P = Kp<:1G, i.e., P~Y, whichcontradiets the choice of P. Hence, K is noninvariant in G and ]K:corGK I is divisible by p. Then in G there is an invariant subgroup S of index p. Again this is a contradiction. Thus, J is the unique maximal sub- group of Gp, i.e., Gp is cyclic. Then by Corollary 4.2.2 of [i] of L. A. Shemetkov we see that G either is p-solvable with p-length less than or equal to 1 or possesses a chief fac- tor which is a simple group of order divisible by IGIp. Since in the chief series of the

group G G ~ J ~ . . . --~ E

446

LITERATURE CITED

I. L. A. Shemetkov, "Concerning partly solvable finite groups," Matem. Sb., 72, No. i, 97-107 (1968).

2. P. Hall and G. Higman, "On p-length of p-solvable groups and reduction theorems for Burnside problem," Proc. London Math. Soc., i, No. 21, 1-42 (1956).

3. S.A. Churnikhin and L. A. Shemetkov, "Finite groups," in: Science: Algebra, Topology, Geometry, VINITI, Moscow (1971), pp. 7-70.

4. B. Huppert, Endliche Gruppen i, Springer, Berlin (1967). 5. L. A. Shemetkov, "On complementation of the F-coradical and the constructions of the

F-hypercenter of a finite group," Dokl. Akad. Nauk BSSR, 13, No. 3, 204-206 (1974). 6. M. P. Hale, "Normally closed saturated formations," Proc. Amer. Math. Soc., 33, No. 2,

337-342 (1972). 7. R. Carter and T. Hawkes, "The F-normalizers of a finite soluble group," J. Algebra, ~,

No. 2, 175-202 (1967). 8. M. V. Sel'kin, "On maximal subgroups of finite groups," Dokl. Akad. Nauk. BSSR, 18,

No. ii, 969-972 (1974). S. A. Chunikhin, Subgroups of Finite Groups [in Russian], Nauka i Tekhnika, Minsk ( 1 9 6 4 ) . W. E. Deskins, "A condition for the solvability of a finite group, III," Illinois J. Math., ~, No. 2, 306-313 (1961).Gp<~G. G. A. Chambers, "On the conjugacy of injectors," Proc. Amer. Math. Soc., 28, No. 2, 358-360 (1971). R. Schmidt, "Modular subgroups of finite groups, II, III," Illinois J. Math., 14, No. 2, 344-362 (1970).

,

I0.

ii.

12.

AN EXAMPLE OF A LOCALLY UNSOLVABLE DIFFERENTIAL EQUATION

OF QUASIPRINCIPAL TYPE WITH A REAL-VALUED WEIGHTED PRINCIPAL SYMBOL

N. A. Shananin UDC 517.9

In this note we show that the equation

_ { ( • 1 7 7 ~ = . ~ o ( ~ 1 ~ , } ~ = s

i s l o c a l l y unso lvab le at the o r i g i n o f the coord ina te system. This equat ion be- longs to the c lass t ha t genera l i zes the p r i n c i p a l type to the case of weighted derivatives. The example is interesting because the weighted principal symbol is real (in this situation, equations of principal type are solvable) but the un- solvability depends on the behavior of the lower-order terms in a neighborhood of the zeros of the weighted principal symbol.

Suppose that in a domain ~cR,~ there is defined a differential operator of order l:

Ox~ . . . . . . oz~,~ '

with infinitely differentiable coefficients. Let (mz, ., mn) be a set of positive in- tegers; in what follows we call it a weight set. We define the weighted order m of the op- erator P(x, Dx) relative to the weight set (mz, ., mn) as the exact upper bound of the

Moscow Institute of Electronic Engineering. Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 755-761, May, 1976. Original article submitted July 24, 1974.

This material is protected by copyright registered in the name o f Plenton Publishing Corporation, 227 West ] 7th Street. New York, N. Y. 10011. No part o f thispublication may ~e reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microJTlming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $ 7.50.

447