RENDICONT1 DEL CIRCOLO MATEMATICO DI PALERMO Serie IL Tomo XLII (1993), pp. 362-368

C O M O N O L I T H I C S U B N O R M A L S U B G R O U P S G E N E R A T I N G A F I N I T E G R O U P

MIGUEL TORRES

Every finite group is generated by minimal sets of subnormal comonolithic subgroups all this sets having the same number of members and whose pcrfcct subgroups being the same for every such a set. The proof of this result depends on some well known results about subnormal subgroups, mainly due to H. Wielandt.

0 Introduction.

A finite group G is called comonolithic (or single-headed) if it has

a unique maximal normal subgroup Mc. In [1] K. Doerk proves that

every finite group G can be generated by its comonolithic subnormal

subgroups and that in the soluble case their maximal normal subgroups

generate a normal subgroup K of G such that G/K is nilpotent with

square-free exponent.

The aim of this short note is to further explore this situation

by giving minimal sets of comonolithic subnormal subgroups that can

generate the group G, in a way that could be considered as a certain

comonolithic version of the Burnside's Basis Theorem.

All groups considered here are finite.

COMONOLITHIC SUBNORMAL SUBGROUPS GENERATING A FINITE GROUP 363

.

We shall very often use the following result that appears as

Theorem 2.2 in [2].

LEMMA 1. If A is a perfect comonolithic subnormal subgroup

of (A1, A2) where Ai is subnormal in (A~, A2) for i = 1,2; then A is

contained in some A,.

The next two points are slight extensions of the statements a)

and b) in the above mentioned result of K. Doerk; no proofs of the

already known facts are included.

LEMMA 2. If M is a maximal normal subgroup of a group G,

then a minimal subnormal supplement S of M in G is comonolithic,

it is maximal with respect to the property of being a comonolithie

subgroup of G and G / M is isomorphic to 5;/Ms. Moreover if G / M is

nonabelian, S is unique.

Proof If S' is contained properly in a comonolithic subgroup T

of G, then S' < MT and G = M T , so we have MT = T A M and

therefore G = S M <_ (T A M ) M = M, a contradiction. In the case G / M

nonabelian the uniqueness of S appears in the proof of Satz 21 in [3].

Let us denote by E the set of all comonolithic subnormal

subgroups of G that are maximal with respect to these properties

and consider it as the disjoint union of Z0, the subset of the perfect

subgroups in Z, and the different E v, formed by those subgroups S in Z

such that S / M s is cyclic of order p. As an easy consequence of lemma

2, two distinct maximal normal subgroups of G with nonabelian factor

groups cannot have the same supplement in E0, therefore we have an

injective map from the set of all maximal normal subgroups M of G

such that G / M is nonabelian into the set Z0.

LEMMA 3. Given E0 = {Al,. . . , At) and Z v as above, we have

i) G = GoG1 where Go = AI...A2, A~ normal in G, and Gl = (BIB E

364 MIGUEL TORRES

uz ).

ii) If K = IMsIS C E), then G/K is isomorphic to the direct product

A1/MA, • ... • A~/MA, • Gift /K-

and GI K/ K is nilpotent with square-free exponent.

Proof i) G is generated by all subgroups in E and due to Satz 20

in [3] all members of Y-~ are normalized by all subgroups in Z, and

therefore they are normal in G.

ii) Obviously G/K = (GoK/tf)(G1K/K) where GoK/tf is the

product of the AiK/K; but AiK/K is isomorphic to A,/A2 N K; on

the other hand Ai (q lC contains JVIA,, hence the factors of GoK/K are

either trivial or simple nonabelian. In the first case Ai N K = A, so

Ai is contained in K, and due to Lemma 1 Ai is contained in some

Ms, S in Z, a contradiction. By Lemma 1 again A2K*Aj K if i*j and

therefore

GoK/K ~- El x ... • Er

where Ei = Ai/MA,.

The structure of GIK/K can be established as in the soluble case

and the other points of the proof are straightforward.

From Lemma 3 we see that G/G1K is isomorphic to the direct

product of E1,...,E,; so the number of maximal normal subgroups

M of G such that G/M is nonabelian is greater than or equal to

the number r of elements in Z0 and so the above mentioned map is

bijective and, in particular, Gl K is the residual of G with respect to the

class E of all completely reducible perfect groups, and what Lemma

3 says is that G is generated modulo this residual G E by Al, ..., A~;

namely

G = AI. . .A~G E.

LEMMA 4. Let N be a normal subgroup of G such that G/N is

a p-group (I9 a prime) having d as its minimal number of generators,

COMONOLITHIC SUBNORMAL SUBGROUPS GENERATING A FINITE GROUP 365

then there exist Bl, . . . , Bd in Zp such that G = (BI,.. . , Bd)N. Moreover n o Bi can be removed.

Proof Let us write r = r the Frattini subgroup of G/N and take maximal subgroups M, of G i = 1, ..., d such that

M I N . . . ~ M d = r

For each index i take Xi the intersection of the subgroups in

this set that are different from M,, so one has M, A Xi = r and it is

easily seen that a minimal subnormal supplement Bi of r in X, is a

minimal subnormal supplement of M, in G; therefore Bi is in Zp. But

G = X I . . . X d -- ( 1 ~ 1 , . . . , Bd)r and hence

G/N = ((BI, ..., Ba)N/N)r

therefore G = (B1, ..., Ba)N.

The minimality of the set of these Bi is due to the Burnside's

Basis Theorem applied to the p-group GIN.

LEMMA 5. Let N and H be normal subgroups of G having coprime indices and M a maximal normal subgroup of G containing

N, then every comonolithic subnormal supplement of M in G is

contained in H.

Let us consider the nilpotent residual G N of a group G and the

different prime divisors Pi i = 1, ..., s of the order of G/G N, then we

have that if Ti/G N is the Sylow p,-subgroup of G/G N, Ti and OPi(G) have coprime indices, so using Lemma 4 one can write

G = (B,1 , . . . , Bid,)OPi(G),

where B O is comonolithic in the set s and d2 is the minimal number

of generators of the p/-residual of G. But by Lemma 5, B O is contained

in Ti so we have

7", = (Bii, ..., Bid~)G N

3 6 6 MIGUEL TORRES

and then

c = ( B l l , ..., ...,

Let us denote by X, the subgroup generated by the subgroups

/3,~ j = 1,...,d, and by X the subgroup generated by these X,. Using

(1.5) Satz and (2.4) Satz in [4], one deduces easily that

o [ ( x , ) : < o p i ( B , , ) , . . . , = =

and by (1.4) in [4] the subgroups X, permute pairwise, so X = XI. . .X~

and G = X G N. But by (1.5) in [4], once more, one can write

GN= XNG NN and then G = X G NN. Finally we reach the soluble

residual G s of G, namely we get G = X G s i.e. the subgroups B v

generate G modulo the soluble residual of G.

With these preliminaries we can establish the main point in this

note.

PROPOSITION. Let G be a group such that

i) d, is the minimal number o f generators o f G/OPi(G), p, 1 < i < s

the different prime divisors o f the order o f G / G N,

and

ii) Z0 = {AI, . . . ,AT}.

Then for ever)' i = 1,..., s there are di subgroups B v in Zp, such

that

G = A , . . . A T < B , , , . . . , B , d , ) . . . <B~I,...,B~d,)

and no A, or B,~ can be removed.

Proof Let us write Xi, X , and G s as above, then we see that

G : X G s : X 1 . . . X ~ G s

and by Satz 22 in [3], the soluble residual is generated by the of

the perfect comonolithic subnormal subgroups of G, obviously we can

consider only the maximal members of this set; they are normalized

by this residual due to Satz 20 in [3]. Take A one of them, if A is

COMONOLITHIC SUBNORMAL SUBGROUPS GENERATING A FINITE GROUP 367

not contained in X, A must be normalized by X and therefore it is

normal in G, so MA is also normal in G and A l M A is a minimal

normal subgroup of G/MA that it is a nonabelian simple group. On the

other hand X M A / M A is a subnormal subgroup of G/MA which clearly

does not contain A l M A (otherwise A would be contained in X M A ,

contradicting Lemma 1); then using Satz (2) in [5] we deduce that

X M A centralizes ALMA. By the same reasons, all perfect comonolithic

generators of the residual G s different from A must centralize A / M A

SO

G s < ACG(A/MA)

This gives us that G = A C c ( A / M A ) , so A is in Z0, and finally

G = A I . . . A ~ X 1 . . . X , .

By Lemma 1 no A, can be removed; so assume now that

G = A1 .. . A~(B12, �9 �9 �9 Big, )X2... Xa

then by [4], Satz (2.4) we have

0 p' (G) = A1 .. . A~O p' (-B12) �9 �9 �9 0 p' ( B l d , ) X 2 . . . X 8

and therefore G = (Bl2, ..., Bzd,)OP'(G), against Lemma 4.

If X and A, are as above and we write M, = MA, and N, = [A~, M,]

we can establish the following

COROLLARY. For every group G, we have

i) G N = A1. . .ArX N

ii) G E = MI. . .MTX

iii) [G E, G N] = Nl ...N~X N

Proof i) This is a consequence of (1.4) in [4].

ii) If we denote H = M1...MT, then the factor group G = G / H

can be expressed as a product

Q = AI...A -X

3 6 8 MIGUEL TORRES

(the bar denotes the epimorphic image) where -A1...-~ is a direct product of r minimal normal subgroups of G, these are simple groups and ,4i A )f, = 1; therefore by [5], Satz 2 fi~i centralizes )f and so )~ is a normal subgroup of G such that G / X is isomorphic to the direct product of the groups fi~i i = 1,..., r. But this implies that G E = HX.

iii) [G N, G E] = [A1 ...A~X N, GE] = [A1, GE]...[Ar, GE][X N, GEl;

but [A~, G E] is contained in Ai as a proper subgroup and hence we have [Ai, G E] is contained in Mi. Using the Three Subgroups Lemma we can deduce that [Ai,G E] is contained in N,; on the other hand X normalizes X N and M, so we have

[X N,G E] = [X N,MI.. .M~X] = [X N,X][X N,M1...M~] =

x N [ x N, MI ... MT] ~_ xN[G E, A1 ... A~] <_ XNN1 ... N~.

Finally [G N, G E] < Nl ...N~. The other inclusion is trivial.

Remark. The commutator subgroup in iii) is precisely the residual of G with respect to the class of the generalized nilpotent groups and so this corollary says that the nilpotent and the generalized nilpotent residuals of a group G are the products of the corresponding residuals of the comonolithic subnormal generators appearing in the Proposition.

REFERENCE

[1] Doerk K., (lber den Rand einer Fittingklasse endlicher aufl6sbarer Gruppen J. of Algebra 51 (1978), 619-630

[2] Pense J. Outer Fitting Pairs J. of Algebra 119 (1988), 34-50. [3] Wielandt H., Eine Verallgemeinerung der invarianten Untergruppen Math. Z.

45 (1939), 209-244.

[4] Wielandt H., Vertauschbare nachinvariante Untergruppen Abh. Math. Sem. Univ. Hamburg 21 (1957), 55-62.

[5] Wielandt H., Uber den normalisator der subnormalen Untergruppen Math. Z. 69 (1958), 463-465.

Pervenuto il 7 settembre 1992.

Departamento de Matematicas Universidad de Zaragoza 50009 Zaragoza (Spain)