Extending the Kegel Wielandt theorem through π-decomposable groups

doi:10.1017/CBO9780511842474.008

Extending the Kegel Wielandt theorem through π-decomposable groups

Published online by Cambridge University Press: 05 July 2011

L. S. Kazarin ,

A. Martínez-Pastor and

Edited by

G. C. Smith and

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

DOI: https://doi.org/10.1017/CBO9780511842474.008 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2011

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References

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Extending the Kegel Wielandt theorem through π-decomposable groups

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