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Generalizations of the Sylow theorem

Published online by Cambridge University Press:  05 July 2011

By
Danila O. Revin  and
Evgeny P. Vdovin
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
Danila O. Revin
Affiliation:
Novosibirsk State University
Evgeny P. Vdovin
Affiliation:
Novosibirsk State University
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

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.

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Groups St Andrews 2009 in Bath , pp. 488 - 519
Publisher: Cambridge University Press
Print publication year: 2011

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