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Extending the Kegel Wielandt theorem through π-decomposable groups

Published online by Cambridge University Press:  05 July 2011

By
L. S. Kazarin ,
A. Martínez-Pastor  and
M. D. Pérez-Ramos
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
L. S. Kazarin
Affiliation:
Yaroslavl P. Demidov State University
A. Martínez-Pastor
Affiliation:
Universidad Politécnica de Valencia
M. D. Pérez-Ramos
Affiliation:
Universitat de València
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

A celebrated theorem of Kegel and Wielandt asserts the solubility of a finite group which is the product of two nilpotent subgroups. In this survey we report on some extensions of this result by considering π-decomposable subgroups, for a set of primes π, instead of nilpotent groups.

Introduction

The study of groups which can be factorised as the product of two subgroups has developed extensively in recent decades. The general aim is to obtain information about the structure of the whole group from the structure of the subgroups in the factorization, and vice versa. An example is the well known theorem of Kegel and Wielandt which establishes the solubility of a finite group factorised as the product of two nilpotent subgroups. This result has been the motivation for a wide variety of results in the literature. In particular some of them consider the situation when either one or both of the factors are π-decomposable, for a set of primes π. This paper is a survey article containing a detailed account of recent achievements which extend the Kegel–Wielandt theorem in this direction.

Only finite groups are considered in this paper.

Let us start with an explicit statement of the starting point of our development:

Theorem 1.1 (Kegel [14] and Wielandt [19])If the group G = AB is the product of two nilpotent subgroups A and B, then G is soluble.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 415 - 423
Publisher: Cambridge University Press
Print publication year: 2011

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References

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