Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-20T06:30:27.375Z Has data issue: false hasContentIssue false
  • English
  • Français

On solvable groups with one vanishing class size

Part of: Representation theory of groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  14 September 2020

M. Bianchi Open the ORCID record for M. Bianchi [Opens in a new window] ,
E. Pacifici Open the ORCID record for E. Pacifici [Opens in a new window] ,
R. D. Camina  and
Mark L. Lewis Open the ORCID record for Mark L. Lewis [Opens in a new window]
M. Bianchi
Affiliation:
Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, via Saldini 50, 20133Milano, Italy ([email protected]; [email protected])
E. Pacifici
Affiliation:
Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, via Saldini 50, 20133Milano, Italy ([email protected]; [email protected])
R. D. Camina
Affiliation:
Fitzwilliam College, Cambridge CB3 0DG, UK ([email protected])
Mark L. Lewis
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Finite groupsvanishing conjugacy classes

MSC classification

Primary: 20E45: Conjugacy classes
Secondary: 20C15: Ordinary representations and characters
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 5 , October 2021 , pp. 1467 - 1486
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Dedicated to Carlo Casolo

References

Ballester-Bolinches, A., Cossey, J. and Li, Y.. Mutually permutable products and conjugacy classes. Monatsh. Math. 170 (2013), 305–310.CrossRefGoogle Scholar
Bianchi, M., Lewis, M. L. and Pacifici, E.. Groups with vanishing class size p. J. Group Theory 23 (2020), 79–83.CrossRefGoogle Scholar
Brough, J.. On vanishing criteria that control finite group structure. J. Algebra 458 (2016), 207–215.CrossRefGoogle Scholar
Camina, A. R. and Camina, R. D.. The influence of conjugacy class sizes on the structure of finite groups a survey, Asian-Eur. J. Math. 4 (2011), 559–588.Google Scholar
Cheng, K. N., Deaconescu, M., Lang, M. L. and Shi, W.. Corrigendum and Addendum to ‘Classification of finite groups with all elements of prime order’. Proc. Amer. Math. Soc. 117 (1993), 1205–1207.CrossRefGoogle Scholar
Dolfi, S. and Jabara, E.. The structure of finite groups of conjugate rank 2. Bull. London Math. Soc. 41 (2009), 916–926.CrossRefGoogle Scholar
Dolfi, S., Navarro, G., Pacifici, E., Sanus, L. and Tiep, P. H.. Non-vanishing elements of finite groups. J. Algebra 323 (2010), 540–545.CrossRefGoogle Scholar
Dolfi, S., Pacifici, E. and Sanus, L.. Groups whose vanishing class sizes are not divisible by a given prime. Arch. Math. 94 (2010), 311–317.CrossRefGoogle Scholar
Dolfi, S., Pacifici, E., Sanus, L. and Spiga, P.. On the vanishing prime graph of solvable groups. J. Group Theory 13 (2010), 189–206.CrossRefGoogle Scholar
Dolfi, S., Pacifici, E. and Sanus, L.. On zeros of characters of finite groups. In Group theory and computation (ed. Sastry, N. S. N. and Yadav, M. K.). (New York: Springer, 2018), 41–58.Google Scholar
Isaacs, I. M.. Character theory of finite groups (San Diego, California: Academic Press, 1976).Google Scholar
Isaacs, I. M.. Groups with many equal class size. Duke Math. J. 37 (1970), 501–506.CrossRefGoogle Scholar
Isaacs, I. M., Navarro, G. and Wolf, T. R.. Finite group elements where no irreducible character vanishes. J. Algebra 222 (1999), 413–423.CrossRefGoogle Scholar
Ishikawa, K.. On finite p-groups which have only two conjugacy lengths. Israel J. Math. 129 (2002), 119–123.CrossRefGoogle Scholar
Itô, N.. On finite groups with given conjugate types, I. Nagoya Math. J. 6 (1953), 17–28.CrossRefGoogle Scholar