Abstract

Let π be a set of primes. Generalizing the properties of Sylow p-subgroups, P.Hall introduced classes Eπ, Cπ, and Dπ of finite groups possessing a π-Hall subgroup, possessing exactly one class of conjugate π-Hall subgroups, and possessing one class of conjugate maximal π-subgroups respectively. In this paper we discuss a description of these classes in terms of a composition and a chief series of a finite group G.

Introduction

In 1872, the Norwegian mathematician L. Sylow proved the following outstanding theorem.

Theorem 1.1 (L. Sylow [76])Let G be a finite group and p a prime. Assume |G| = pαm and (p, m) = 1. Then the following statements hold:

1. (E) G possesses a subgroup of order pα (the, so-called, Sylow p-subgroup);

2. (C) every two Sylow p-subgroups of G are conjugate;

3. (D) every p-subgroup of G is included in a Sylow p-subgroup.

A natural generalization of the concept of Sylow p-subgroups is the notion of π-Hall subgroups. We recall the definitions. Let G be a finite group and π be a set of primes. We denote by π′ the set of all primes not in π, by π(n) the set of all prime divisors of a positive integer n and for a finite group G we denote π(|G|) by π(G). A positive integer n with π(n) ⊆ π is called a π-number and a group G with π(G) ⊆ π is called a π-group.