MATHEMATICS

L. A. SHEMETKOV

ON EMBEDDING THEOREMS AND MAXIMAL SUBGROUPS OF FINITE GROUPS

(Presented by Academician A. I. Mal’cev, 28 V 1962)

§ 1. The main results of the present note are embedding theorems for subgroups of finite groups, connected with not necessarily Hall subgroups (by a Hall subgroup we mean a subgroup whose order is relatively prime to its index).

Theorem 1 generalizes Theorem 15 of Tibilletti (¹); Theorem 3 generalizes a theorem of H. Wielandt (²); Theorems 7–11 generalize the results of the work of S. A. Rusakov (³); Theorem 13 generalizes Theorems 2 and 3 of the work of S. A. Chunikhin (⁴); Theorem 14 generalizes Hall’s Theorem D5 (⁵). In § 6 analogues are given of a theorem of H. Wielandt (²) connected with the “isomorphic” embedding of subgroups; in § 7 the influence, in a special case, of properties of maximal subgroups on the properties of the group is considered; in § 8 Iwasawa’s theorem on finite (J)-groups (⁶) is generalized; in § 9 “(\Pi)-theorems” corresponding to Theorems 22–24 of the work (⁷) are given.

§ 2. Let (\Pi) be some (empty or nonempty) set of primes; (\mathfrak{G}) a finite group of order ((\mathfrak{G}) = g = mn), where (m \geqslant 1) is the greatest (\Pi)-divisor (⁸) of the order (g); if (\mathfrak{G}) has at least one subgroup of order (m), then by (\mathfrak{G}{\Pi}) we shall denote some subgroup of order (m) of the group (\mathfrak{G}); for (m > 1) put (p) equal to the greatest prime divisor of the number (m), and for (m = 1) put (p = 1); (\mathfrak{E}) is the identity subgroup of the group (\mathfrak{G}); a (\Pi d)-group is a group whose order is divisible by some prime from (\Pi); (h) is the greatest (\Pi')-divisor of the natural number (h) ((\Pi') is the set of all primes not belonging to (\Pi));
[
h = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k},\quad k \geqslant 1,
]
is the canonical decomposition of the natural number (h > 1).

Let (\rho) denote the set of all primes with some ordering introduced on it. A group (\mathfrak{H}) of order
[
h = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k},\quad k \geqslant 1,
]
will be called strongly (\rho)-dispersive if the order of the normalizer in (\mathfrak{H}) of each of its (p_i)-subgroups ((i = 1,2,\ldots,k)) is divisible by ((\mathfrak{G})_{\Pi_i}), where (\Pi_i) is the set of all those primes which follow (p_i) in the given ordering. Obviously, a strongly (\rho)-dispersive group is (\rho)-dispersive (⁹). We shall regard (\mathfrak{E}) as a trivially strongly (\rho)-dispersive group.

The notions of (p)-speciality, (\Pi)-solvability, (\Pi)-separability, and strong (\Pi)-solvability are taken by us from the work (¹⁰). For groups of type (S), see (¹¹,¹²). We also use the notion of the indexial of a finite group and the definitions and notation connected with it from the work (¹³).

In what follows, by (\sigma) and (\tau) we shall denote such (empty or nonempty) sets of primes that (\sigma \cap \tau) is empty and (\Pi = \sigma \cup \tau). The notation (\mathfrak{G}{\sigma}), (\mathfrak{G}}), (\mathfrak{H{\sigma}), and so on, is defined analogously to (\mathfrak{G}).

§ 3. Let (h) be some divisor of the order of the group (\mathfrak{G}). We introduce for consideration the following properties of finite groups: (E(h))—in (\mathfrak{G}) there is at least one subgroup (\mathfrak{H}) of order (h); (C(h))—(\mathfrak{G}) has property (E(h)) and any two subgroups of order (h) of the group (\mathfrak{G}) are conjugate in (\mathfrak{G}); (D(h))—(\mathfrak{G}) has property (C(h)), and every subgroup of order dividing (h), of the group ...

if (\mathfrak G) is contained in some subgroup of order (h) of the group (\mathfrak G); (D^s(h))—(\mathfrak G) has property (D(h)) and its subgroups of order (h) are soluble; (D^{ss}(h))—(\mathfrak G) has property (D(h)) and its subgroups of order (h) are supersoluble.

If (h=m), then instead of (E(h), C(h), D(h), D^s(h)), and (D^{ss}(h)) we shall use, respectively, the symbols (E_\Pi, C_\Pi, D_\Pi, D_\Pi^s), and (D_\Pi^{ss}). If (\mathfrak A) is a subgroup of order (a) of the group (\mathfrak G), then, along with the symbols (C(a), D^s(a)), etc., we shall also use the symbols (C(\mathfrak A), D^s(\mathfrak A)), etc.

We shall say that the strong (D(h))-theorem holds for the group (\mathfrak G) if (\mathfrak G) has property (D(\mathfrak L)) for each of its subgroups (\mathfrak L) whose order divides (h). If (h=m), then we shall speak of the strong (D_\Pi)-theorem ((^3)).

§ 4. A divisor (h=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}), (k\geqslant 1), of the order of the group (\mathfrak G) will be called a quasi-Hall divisor of the order of the group (\mathfrak G) if, for each (i=1,2,\ldots,k), the following condition holds: the normalizer (\mathfrak N) of every subgroup (\mathfrak A) of (\mathfrak G) such that ((\mathfrak A)) divides (h/p_i^{\alpha_i}) and (p_i^{\alpha_i}) divides ((\mathfrak N)), has property (C(p_i^{\alpha_i})). We shall also regard the identity as a quasi-Hall divisor of the order of (\mathfrak G).

A subgroup (\mathfrak H) of the group (\mathfrak G) will be called a quasi-Hall subgroup of the group (\mathfrak G) if ((\mathfrak H)) is a quasi-Hall divisor of the order of (\mathfrak G). It is easy to see that every Hall subgroup of the group (\mathfrak G) is also its quasi-Hall subgroup.

Theorem 1. Let (\mathfrak G) have a quasi-Hall (\rho)-dispersive subgroup (\mathfrak H) of order (h), and let (\mathfrak L) be a (\rho)-dispersive subgroup of order (l), dividing (h), of the group (\mathfrak G), with ((l,h/l)=1). Then there is an element (G\in\mathfrak G) such that (\mathfrak L^G\supseteq\mathfrak H).

Theorem 2. Let (\mathfrak G) have a quasi-Hall strongly (\rho)-dispersive subgroup (\mathfrak H). Then for every (\rho)-dispersive subgroup (\mathfrak L) of order dividing ((\mathfrak H)) of the group (\mathfrak G), there is an element (G\in\mathfrak G) such that (\mathfrak L^G\subseteq\mathfrak H).

Theorem 3. Let (\mathfrak G) have a quasi-Hall subgroup (\mathfrak H=\mathfrak H_\sigma\times\mathfrak H_\tau), where (\mathfrak H_\sigma) is special, and suppose (\mathfrak G) has property (D(\mathfrak H_\tau)). Then (\mathfrak G) has property (D(\mathfrak H)).

Theorem 4. Let (\mathfrak G) have a quasi-Hall subgroup (\mathfrak H=\mathfrak H_\sigma\times\mathfrak H_\tau), where (\mathfrak H_\sigma) is strongly (\rho)-dispersive, and suppose (\mathfrak G) has properties (D(\mathfrak H_\sigma)) and (D(\mathfrak H_\tau)). Then (\mathfrak G) has property (D(\mathfrak H)).

Theorem 5. Let (\mathfrak G) have a subgroup (\mathfrak G_\Pi=\mathfrak G_\sigma\times\mathfrak G_\tau), where (\mathfrak G_\sigma) is strongly (\rho)-dispersive, and suppose (\mathfrak G) has properties (D_\sigma) and (D_\tau). Then (\mathfrak G) has property (D_\Pi).

Theorem 6. Let (\mathfrak G) have a subgroup (\mathfrak H=\mathfrak H_\sigma\times\mathfrak G_\tau), where (\mathfrak H_\sigma) is (\rho)-dispersive. Suppose that (\mathfrak G) and all its subgroups containing (\mathfrak H_\sigma) have property (D(\mathfrak H_\sigma)), and suppose that (\mathfrak G) has property (D_\tau). Then (\mathfrak G) has property (D(\mathfrak H)).

Theorem 7. Let (h=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}), (k\geqslant 1), be a quasi-Hall divisor of the order of (\mathfrak G), and suppose that, for each (i=1,2,\ldots,k), (\mathfrak G) has a cyclic subgroup of order (p_i^{\alpha_i}). Then the strong (D(h))-theorem holds for (\mathfrak G).

Theorem 8. Let (\mathfrak G) have a quasi-Hall subgroup (\mathfrak H=\mathfrak H_\sigma\times\mathfrak H_\tau), where all Sylow subgroups of (\mathfrak H_\sigma) are cyclic, and suppose (\mathfrak G) has property (D(\mathfrak H_\tau)). Then (\mathfrak G) has property (D(\mathfrak H)).

Theorem 9. Suppose that all Sylow subgroups of the group (\mathfrak G) corresponding to primes in (\sigma) are cyclic. Let (\mathfrak G) have a subgroup (\mathfrak H=\mathfrak H_\sigma\times\mathfrak G_\tau), and suppose (\mathfrak G) has property (D_\tau). Then (\mathfrak G) has property (D(\mathfrak H)).

Theorem 10. Let (\mathfrak G) have a quasi-Hall subgroup (\mathfrak H=\mathfrak H_\sigma\times\mathfrak H_\tau), where all Sylow subgroups of (\mathfrak H_\sigma) are cyclic, and suppose that the strong (D(\mathfrak H_\tau))-theorem holds for (\mathfrak G). Then the strong (D(\mathfrak H))-theorem holds for (\mathfrak G).

Theorem 11. Let all Sylow subgroups of the group (\mathfrak G) corresponding to the prime numbers in (\sigma) be cyclic. Let (\mathfrak G) have a subgroup (\mathfrak H=\mathfrak H_\sigma\times \mathfrak G_\tau), and let the strong (D_\tau)-theorem hold for (\mathfrak G). Then the strong (D(\mathfrak H))-theorem holds for (\mathfrak G).

§ 5. A divisor (h=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}), (k\geq 1), of the order of a group (\mathfrak G) such that (\mathfrak G) has the property (C(p_i^{\alpha_i})) for each (i=1,2,\ldots,k), and also (1), will be called a quasi-Sylow divisor of the order of the group (\mathfrak G).

A divisor (h) of the order of the group (\mathfrak G) such that ((h,l)) is a quasi-Sylow divisor of the order (l) of an arbitrary characteristic subgroup (\Omega) of the group (\mathfrak G) (in particular, (\Omega=\mathfrak G)) will be called a strong quasi-Sylow divisor of the order of the group (\mathfrak G).

We shall call the indexed set ((h){R,f}) of the group (\mathfrak G) Hall if: 1) ((h)/\mathfrak G_i), (i=\beta+1,\beta+2,\ldots,\nu).}) is a refined indexed set of the group (\mathfrak G); 2) (\mathfrak G_{\beta-1}/\mathfrak G_\beta) has the property (D^s(f_\beta)); 3) (\mathfrak F_i/\mathfrak G_i) is a special Hall subgroup of the group (\mathfrak G_{i-1

Theorem 12. Let (h) be such a quasi-Sylow divisor of the order of a (\Pi)-solvable group (\mathfrak G) that (h_{\Pi'}) is equal either to (1) or to (n). Then (\mathfrak G) has the property (D(h)).

Theorem 13. Let (h) be such a strong quasi-Sylow divisor of the order of a (\Pi)-solvable group (\mathfrak G) that (h_{\Pi'}) is equal either to (1) or to (n). Then (\mathfrak G) has the property (D(h)).

Theorem 14. Let ((h)_{R,f}) be a Hall indexed set of the group (\mathfrak G), and let (h) be a quasi-Sylow divisor of the order of (\mathfrak G). Then (\mathfrak G) has the property (D^s(h)).

§ 6. We shall say that a group (\mathfrak G) has the property (I(h)) if (\mathfrak G) has at least one subgroup (\mathfrak H) of order (h), and for any subgroup (\Omega) of the group (\mathfrak G) whose order divides (h), there is a subgroup (\Omega^) of (\mathfrak H) such that (\Omega) and (\Omega^) are isomorphic. We shall say that the strong (I(h))-theorem holds for (\mathfrak G) if (\mathfrak G) has the property (I(l)) for each of its subgroups (\Omega) whose order (l) divides (h). If (h=m), then instead of (I(h)) we shall use the symbol (I_\Pi). The notations (I_\sigma) and (I_\tau) are introduced analogously. If (\mathfrak A) is a subgroup of order (a) of the group (\mathfrak G), then along with (I(a)) we shall also use the symbol (I(\mathfrak A)).

Theorem 15. Let (\mathfrak G) have a quasi-Hall subgroup (\mathfrak H=\mathfrak H_\sigma\times \mathfrak H_\tau), where (\mathfrak H_\tau) is (\rho)-dispersive, and let (\mathfrak G) have the properties (I(\mathfrak H_\sigma)) and (I(\mathfrak H_\tau)). Then (\mathfrak G) has the property (I(\mathfrak H)).

Theorem 16. Let (\mathfrak G) have a quasi-Hall subgroup (\mathfrak H=\mathfrak H_\sigma\times\mathfrak H_\tau), where (\mathfrak H_\sigma) is (\rho)-dispersive, and let the strong (I(\mathfrak H_\sigma))-theorem and the strong (I(\mathfrak H_\tau))-theorem hold for (\mathfrak G). Then the strong (I(\mathfrak H))-theorem holds for (\mathfrak G).

§ 7. If in the group (\mathfrak G) the set of all maximal subgroups with core (\mathfrak E) ((^9)) is nonempty, then this set is divided into classes by the isomorphy relation ((^{14})). The classes obtained will be called classes of isomorphic maximal subgroups with core (\mathfrak E). We shall say that (\mathfrak G) is a group with one class of isomorphic maximal subgroups with core (\mathfrak E) if (\mathfrak G) has only one class of isomorphic maximal subgroups with core (\mathfrak E).

Theorem 17. Let (\mathfrak G) be a group with one class of isomorphic maximal subgroups with core (\mathfrak E). If among the isomorphic subgroups of the given class there is at least one subgroup possessing the property (E_\Pi^n) ((^5)), then (\mathfrak G) has the property (D_\Pi^s).

Theorem 18. Let (\mathfrak G) be a group with one class of isomorphic maximal subgroups with core (\mathfrak E). If among the isomorphic subgroups of the given class there is at least one (\Pi)-solvable or (\Pi)-separable subgroup, then (\mathfrak G) is respectively (\Pi)-solvable or (\Pi)-separable.

§ 8. If (\mathfrak G\ne\mathfrak E), then a series of subgroups

[
\mathfrak G=\mathfrak H_0\supset \mathfrak H_1\supset\cdots\supset \mathfrak H_t=\mathfrak E,\qquad t\geq 1,
]

will be called a maximal series of the group (\mathfrak G), if each term of this series

(\mathfrak H_i) ((i=1,2,\ldots,t)) is a maximal subgroup of the preceding member of the series (\mathfrak H_{i-1}). If the group (\mathfrak G=\mathfrak E), then its only maximal series will be considered to be (\mathfrak E,\mathfrak E). If in every maximal series of the group (\mathfrak G) the number of indices, all prime divisors of which belong to (\Pi), is the same, then (\mathfrak G) will be called a (\Pi J)-group.

Theorem 19. If (\mathfrak G) is strongly (\Pi)-solvable, then it is a (\Pi J)-group.

Theorem 20. If (\mathfrak G) is a (\Pi J)-group and ((p!,n)=1), then (\mathfrak G) is strongly (\Pi)-solvable.

Theorem 21. If a (\Pi J)-group (\mathfrak G) has property (E_{\Pi}), then it also has property (D_{\Pi}^{ss}).

§ 9. In the present paragraph (\mathfrak G) will denote a group for which (m>1) and (n>1). A subgroup (\mathfrak H) of the group (\mathfrak G) will be called an (r)-th (\Pi)-maximal subgroup of the group (\mathfrak G), if there exists a maximal series of the group (\mathfrak G)

[
\mathfrak G=\mathfrak H_0 \supset \mathfrak H_1 \supset \cdots \supset \mathfrak H_r=\mathfrak H \supset \cdots \supset \mathfrak H_t=\mathfrak E,
]

where (\mathfrak H_i) is a (\Pi d)-subgroup for every (i=1,2,\ldots,r-1).

Theorem 22. If all proper subgroups of the group (\mathfrak G) are strongly (\Pi)-solvable and ((p!,n)=1), then (\mathfrak G) is either strongly (\Pi)-solvable or a (p)-special group of type (S).

Theorem 23. If in a nonspecial group (\mathfrak G) all its second (\Pi)-maximal subgroups are invariant, then (\mathfrak G) is either a strongly (p)-solvable group of type (S), or the direct product of a group of order (m=p) by a group of type (S).

Theorem 24. If in a group (\mathfrak G) all its third (\Pi)-maximal subgroups are invariant and ((p!,n)=1), then (\mathfrak G) is either strongly (\Pi)-solvable or a (p)-special group of type (S).

In conclusion I express my deep gratitude to S. A. Chunikhin for valuable advice and recommendations.

Gomel Branch
of the Institute of Mathematics and Computational Technology
of the Academy of Sciences of the BSSR

Received
20 V 1962

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