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Pronormal subgroups and transitivity of some subgroup properties

Published online by Cambridge University Press:  05 July 2011

By
Leonid A. Kurdachenko ,
Javier Otal  and
Igor Ya. Subbotin
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
Leonid A. Kurdachenko
Affiliation:
National University of Dnepropetrovsk
Javier Otal
Affiliation:
University of Zaragoza
Igor Ya. Subbotin
Affiliation:
National University, Los Angeles
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 subgroup H of a group G is called pronormal in G if for each element g ∈ G the subgroups H and Hg are conjugate in 〈H, Hg〉. Pronormal subgroups have been introduced by P. Hall, and they play an important role in many studies dedicated to normal structure and Sylow theory of finite and infinite groups and in investigations of arrangement of subgroups in infinite linear groups over rings. Many interesting and important developments have been lately completed in this area by different authors. Thanks to these results, we can see that pronormal subgroups and some other types of subgroups related to them (such as contranormal, abnormal, polynormal, paranormal, permutable subgroups, and so on) are very closely connected to transitivity of some group properties (such as normality, permutability and other) and to (locally) nilpotency of a group. In the current survey, we try to reflect some important new results in this area.

Introduction

Recall that a subgroup H of a group G is called pronormal in G if for each element g ∈ G the subgroups H and Hg are conjugate in 〈H, Hg〉. Pronormal subgroups have been introduced by P. Hall. Important examples of pronormal subgroups are the Sylow p-subgroups of finite groups, the Sylow π-subgroups of finite soluble groups, the Carter subgroups of finite soluble groups and many others. Finite groups with all pronormal subgroups have been described by T.A. Peng in [41, 42].

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Groups St Andrews 2009 in Bath , pp. 448 - 460
Publisher: Cambridge University Press
Print publication year: 2011

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