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On solvable groups with one vanishing class size

Accepted version
Peer-reviewed

Type

Article

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Authors

Bianchi, M 
Pacifici, E 
Camina, RD 
Lewis, ML 

Abstract

Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh. Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79-83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free.

Description

Keywords

Finite groups, vanishing conjugacy classes

Journal Title

Proceedings of the Royal Society of Edinburgh Section A Mathematics

Conference Name

Journal ISSN

0308-2105
1473-7124

Volume Title

0

Publisher

Cambridge University Press

Rights

All rights reserved