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On solvable groups with one vanishing class size

Published online by Cambridge University Press:  14 September 2020

M. Bianchi
Affiliation:
Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, via Saldini 50, 20133Milano, Italy (mariagrazia.bianchi@unimi.it; emanuele.pacifici@unimi.it)
E. Pacifici
Affiliation:
Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, via Saldini 50, 20133Milano, Italy (mariagrazia.bianchi@unimi.it; emanuele.pacifici@unimi.it)
R. D. Camina
Affiliation:
Fitzwilliam College, Cambridge CB3 0DG, UK (rdc26@dpmms.cam.ac.uk)
Mark L. Lewis
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242, USA (lewis@math.kent.edu)

Abstract

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.

Type
Research Article
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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On solvable groups with one vanishing class size
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