Generalizations of the Sylow theorem

Abstract

Abstract Let π be a set of primes. Generalizing the properties of Sylow p -subgroups, P.Hall introduced classes E π , C π , and D π 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| = p α m and (p, m) = 1. Then the following statements hold: (E) G possesses a subgroup of order p α (the, so-called, Sylow p-subgroup); (C) every two Sylow p-subgroups of G are conjugate; (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.