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Structure of finite groups having few conjugacy class sizes

Published online by Cambridge University Press:  05 July 2011

By
Antonio Beltrán  and
María José Felipe
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
Antonio Beltrán
Affiliation:
Universitat Jaume I, Spain
María José Felipe
Affiliation:
Universidad Politécnica de Valencia, Spain
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

The structure of finite groups has a significant influence on the conjugacy class sizes and reciprocally is also influenced by them. In this paper we present some classic and recent contributions which have been obtained during the last forty years related to the structure and properties of those groups having few conjugacy class sizes.

Introduction

The structure of a finite group strongly controls and at the same time is controlled by the sizes of its conjugacy classes and this relation also occurs for the degrees of its irreducible characters. If G is a finite group, we will denote by cs(G) the set of the conjugacy class sizes of G and by cd(G) the set of its irreducible character degrees. It is accepted that there exists certain parallelism between the results accomplished on the group structure from its character degrees and the ones obtained from its class sizes, although the techniques employed to show them may be completely different. We focus our attention to the cardinals of these sets. The fact that they are small for a group may provide a lot of information about it, but we observe that apparently there seems to be no relation between them. For instance, we have that G = SL(2, 5) satisfies |cs(G)| = 4 and |cd(G)| = 6, and that G = A7 satisfies |cs(G)| = 8 and |cd(G)| = 7, whereas the equality holds for G = A5 since |cs(G)| = |cd(G)| = 4.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 124 - 132
Publisher: Cambridge University Press
Print publication year: 2011

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