MATHEMATICS

Ya. G. BERKOVICH

NORMAL DIVISORS OF A FINITE GROUP

(Presented by Academician V. M. Glushkov on 24 I 1968)

1. In group theory it is important not only to establish the existence of a normal divisor, but also its structure and position in the group. In this note a number of results related to this circle of questions are formulated. Only finite groups are considered.

In §2 the discussion concerns (p)-groups. Theorem 1 gives a description of a (p)-group (G) with cyclic Frattini subgroup (\Phi(G)). It is easy to see that this class includes (p)-groups in which all abelian characteristic subgroups are cyclic, so that F. Hall’s theorem on such groups (a shorter, more elementary proof of it was published by D. Gorenstein ((^{1}))) is a special case of Theorem 1. We note that the class of groups considered in Theorem 1 is hereditary for subgroups and homomorphic images, which cannot be said of the class of groups described by F. Hall’s theorem. A corollary of Theorem 2 refines a theorem of C. Hobby ((^{2})). Theorem 3 contains the formulation of a series of new theorems on the number of subgroups of a given order and a given structure in a nonmetacyclic (p)-group, (p>3). Known theorems of G. Miller and M. Tazawa are special cases of Theorem 3. From this theorem two corollaries on normal divisors in a (p)-group are derived (these corollaries in form resemble N. Blackburn’s well-known theorem on (p)-groups without normal divisors of type ((p,p,p))). We note that for regular groups Theorem 3 is true without restrictions on (p). Theorem 4 gives examples of 2-groups (G) for which (\operatorname{Aut}(G)) are 2-groups.

In §3 several results are formulated on groups representable as a product of pairwise permutable subgroups. In Theorem 11, (\Gamma_1^p)-quasinilpotent groups are studied; a complete description is given of solvable groups with this property. In particular, it turns out that (\Gamma_1^2)-quasinilpotent groups are always solvable, while the question of solvability of (\Gamma_1^p)-quasinilpotent groups, if (p) is odd, remains open.

Notation and definitions. (H^G) is the normal closure of the subgroup (H) in the group (G); (\pi) is the set of prime numbers; (\Gamma_1) is the set of all maximal subgroups of the group (G); (\Gamma_1^p) is the set of all those elements of (\Gamma_1) whose order is divisible by (p) (in the case when the set (\Gamma_1^p) is under discussion, we always assume that the order of the group is divisible by (p)); (N(H)) is the normalizer, and (C(H)) is the centralizer of the complex (H) in (G); two subgroups (F) and (H) form a nilpotent pair if they are either incident, or

[
N(F\cap H)\cap F\ne F\cap H\ne N(F\cap H)\cap H;
]

if (M) is a set of subgroups in (G), then we shall call the latter (M)-quasinilpotent if any two elements of (M) form a nilpotent pair; a (p)-group is called extraspecial if its center, commutator subgroup, and Frattini subgroup coincide and have order (p) (for convenience, the identity group is also counted as extraspecial); (Z(G)) is the center of the group (G); (G') is the commutator subgroup of the group (G).

2. Theorem 1. Let the Frattini subgroup of a nonabelian (p)-group (G) be cyclic, and let (\Phi_0) be a subgroup of order (p) in (\Phi(G)). Then (G=AECM), where

in addition: (a) the subgroup (A) is characteristic in (G), abelian, and has exponent (p); (b) the subgroup (E) is extraspecial, and, if (p) is odd, the exponent of (E) does not exceed (p); (c) (C=Z(EC)) is cyclic; (d) the subgroup (AEC) is generated by (\Phi_0) and all subgroups containing (\Phi_0) and invariant in (G) of type ((p,p)) (in particular, (AEC/\Phi_0) has exponent not exceeding (p), and lies in (Z(G/\Phi_0))); (e) the subgroup (M) is cyclic or a 2-group of maximal class, and (\Phi(G)\subseteq M); (f) if (M) is cyclic, then (\Phi(G)\subseteq Z(G)) and (G') has order (p); (g) if (M) is of maximal class, then (G'=\Phi(G)) and (|G:C(\Phi(G))|=2,\ |M|=4|\Phi(G)|); (h) if (A) is cyclic, then (C\cdot\Phi(G)) is an abelian characteristic subgroup of (G).

In particular, if the (p)-group (G) contains no noncyclic characteristic abelian subgroups, then (\Phi(G)) is cyclic, (A\subseteq\Phi_0,\ C\subseteq M), and (G=EM); this is precisely the theorem of P. Hall ((^1)).

Theorem 2. Let a (p)-group (G) be contained as an invariant subgroup in an arbitrary group (H); let (\Phi_H(G)) be the intersection of all (H)-admissible maximal subgroups in (G); and let an (H)-admissible subgroup (N) be supersolubly embedded in (H) and lie in (\Phi_H(G)). If (N) is generated by two elements, then it is metacyclic, provided (N-\Phi(G)) contains no elements of order 4 (the intersection of the empty set of subgroups of (G) is taken to be (G)).

Corollary 1. Let a subgroup (N) be generated by two elements and be invariant in the (p)-group (G). If (N\subseteq\Phi(G)), then (N) is metacyclic.

Theorem 3. Let (G) be a nonmetacyclic (p)-group, (p>3). Then the following assertions hold:

(a) The number of subgroups of order (p^n), (n>2), contained in (G) and metacyclic is divisible by (p).

(b) The number of cyclic subgroups of order (p^n,\ n>1), contained in (G), is divisible by (p^2).

(c) (G) contains a number divisible by (p) of such subgroups of order (p^n,\ n>2), each of which possesses a cyclic subgroup of index (p^2).

(d) The number of subgroups of order (p) contained in (G) is congruent to (1+p+p^2) modulo (p^3).

Corollary 2. If (p>3), then an invariant subgroup (M) in a (p)-group (G) is metacyclic if and only if it contains no subgroup invariant in (G) of order (p^3) and exponent (p).

Corollary 3. If (p>3), then the Frattini subgroup of a (p)-group (G) is metacyclic if and only if (\Phi(G)) contains no subgroup invariant in (G) of type ((p,p,p)).

Theorem 4. Let (P_i=G_i\times C), where the subgroup (C) is a cyclic 2-group, (G_1) is a 2-group of maximal class, and

[
G_2=\operatorname{gp}\langle a,b\mid a^2=b^{2^{\,n-1}}=1,\ aba=b^{1+2^{\,n-2}},\ n>3\rangle .
]

If (G_1) is not a quaternion group, then (\operatorname{Aut}(P_i)) is a 2-group. Moreover, if (P_2) has odd index in a finite group (G), then (G) has an invariant 2-complement.

Theorem 5. Let a group (G) have odd order (p^{3m}), (m>3). Then the number of nonabelian subgroups of order (p^3) contained in (G) is divisible by (p).

For 2-groups, a stronger result has been proved in ((^4)) (if (G) is not a group of maximal class, then the number indicated in the theorem is even divisible by 4).

In the proof of Theorem 4, Frobenius’ theorem on an invariant (p)-complement is used essentially, together with the theorem of Grün—Hall.

1. Theorem 6. Let a soluble group (G=A\cdot B_1\cdot\ldots\cdot B_n), where the subgroups (B_i) are nilpotent and (A) is a (p)-closed group for all (p) dividing (|B_1\cdot\ldots\cdot B_n|). If (A\ne G), then the normal closure of one of the subgroups (A,B_1,\ldots,B_n) is distinct from (G).

This theorem generalizes Kegel’s well-known result on normal subgroups of sets in the product of two nilpotent groups.

A (Z_\pi)-group is a group with cyclic Sylow (p)-subgroups for all (p) in (\pi). A group is called (\pi)-decomposable if its nilpotent (\pi)-Hall subgroup is a direct factor.

Theorem 7. Let (|G|=mn), where (m) is the greatest (\pi)-divisor of (|G|), and for all (p) in (\pi) we have ((p-1,n)=1). If (G) is the product of two (\pi)-decomposable (Z_\pi)-groups, then it contains an invariant subgroup of order (n).

The proof of this theorem is based on the use of results of V. D. Mazurov ((^3)) and of the author ((^4)) on automorphisms of the product of two cyclic (2)-groups.

Theorem 8. Let (G=(P_1\times L_1)\cdot(P_2\times L_2)), where (P_i) is a Sylow (2)-subgroup in (P_i\times L_i). If the (P_i) are metacyclic, then either (G) is solvable, or (G) contains a subgroup whose homomorphic image is (PSL(2,p)), where (p) is a Mersenne prime.

The core (H_G) of a subgroup (H) in a group (G) is the intersection of all subgroups conjugate to (H) in (G). For the definition of a (\pi\varphi)-dispersive group, see ((^5)).

Theorem 9. Let all maximal subgroups of a group (G) that have a given core (H_G) be (\pi\varphi)-dispersive. Then either (G) is a (\pi)-solvable group with (\pi)-length not exceeding (2), or (2) is contained in (\pi), and the (unique) non-(\pi)-solvable chief factor group of (G) contains a subgroup whose homomorphic image is the symmetric group of degree four.

Theorems 10 and 11 were proved jointly by S. L. Gramm and the author.

Theorem 10. Let the set of prime divisors of the index of a maximal non-nilpotent solvable subgroup (H) in (G) be independent of (H) (it is, of course, assumed that (G) contains at least one non-nilpotent solvable subgroup). Then one of the following assertions holds:

(a) (G=P\times L), where (P) is a Sylow subgroup in (G), and (L) is a Schmidt group.

(b) (|G|) contains exactly two distinct prime numbers, and in the factor group of (G) modulo the hypercenter one of the Sylow subgroups is maximal.

Theorem 11. Let (G) be a (\Gamma_1^p)-quasinilpotent group of order divisible by (p). Then:

(a) If (G) is solvable, then: (1a) any two elements of (\Gamma_1^p) are conjugate in (G); (2a) the order of the intersection of any two distinct elements of (\Gamma_1^p) is not divisible by (p); (3a) (G/\Phi(G)) is an elementary group generated by two elements of order (p).

(b) If (G) is solvable, then it is either (\Gamma_1)-quasinilpotent, or is an extension of a minimal invariant (p)-subgroup by means of a (\Gamma_1)-quasinilpotent group of order not divisible by (p).

In particular, from part (a) of the theorem it follows that (\Gamma_1^2)-quasinilpotent groups are solvable, while the question of solvability of (\Gamma_1^p)-quasinilpotent groups for (p>2) remains open. The theorem also easily yields the solvability of a (\pi)-separable (\Gamma_1^p)-quasinilpotent group in the case where (|G|) is divisible by (p) and at least two prime numbers from (\pi) divide (|G|).

Theorem 12. A group (G) is (\pi)-solvable if and only if it satisfies the following conditions:

(a) If (M) is an arbitrary normal divisor in (G), then in it there is a (\pi)-Hall subgroup (P) such that (M\cdot N(P)=G).

(b) If (H) is maximal in (G) and its index in (G) is a (\pi)-number, then for all maximal subgroups in (G) having core (H_G), there is a prime number dividing their indices in (G).

From this theorem one obtains the following necessary and sufficient condition for the conjugacy of maximal subgroups of a (\pi)-solvable group:

Let (F) and (H) be maximal subgroups of a (\pi)-solvable group (G), and suppose that the index of (F) in (G) is a (\pi)-number. Then (F) and (H) are conjugate in (G) if and only if their Sylow subgroups corresponding to the same prime numbers are conjugate in (G).

The results of § 2 of this note were reported by the author at the Ninth All-Union Algebraic Colloquium.

Rostov Civil Engineering
Institute

Received
13 I 1968

REFERENCES

({}^{1}) D. Gorenstein, Pacific J. Math., 19, 1, 77 (1966). ({}^{2}) C. Hobby, Illinois J. Math., 5, 225 (1961). ({}^{3}) V. D. Mazurov, Sibirsk. matem. zhurn., 8, 5 (1967). ({}^{4}) Ya. G. Berkovich, DAN, 179, No. 1, 13 (1968). ({}^{5}) Ya. G. Berkovich, Sibirsk. matem. zhurn., 5, 1, 14 (1964).