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Commuting conjugacy classes: An application of Hall's marriage theorem to group theory

Lookup NU author(s): Dr John Britnell

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Abstract

Let G be a finite group. Define a relation ∼ on the conjugacy classes of G by setting C ∼ D if there are representatives c ε C and d ε D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ∼. One of the proofs involves an application of the famous marriage theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ∼ in the particular cases of symmetric and general linear groups, and by mentioning an open question related to Frobenius groups. © 2009 de Gruyter.

Publication metadata

Author(s): Britnell JR, Wildon M

Publication type: Article

Publication status: Published

Journal: Journal of Group Theory

Year: 2009

Volume: 12

Issue: 6

Pages: 795-802

Print publication date: 01/11/2009

Online publication date: 15/06/2009

ISSN (print): 1433-5883

ISSN (electronic): 1435-4446

Publisher: Walter de Gruyter GmbH

URL: https://doi.org/10.1515/JGT.2009.013

DOI: 10.1515/JGT.2009.013

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