On solvable groups with one vanishing class size. Issue 5 (October 2021)

• Record Type: Journal Article
• Authors: Bianchi, M.; Pacifici, E.; Camina, R. D.; Lewis, Mark L.
• Is Part Of: Proceedings. Volume 151:Issue 5(2021)
• Page Start: 1467
• Page End: 1486
• Publication Date: 2021-10
• Subjects: Finite groups -- vanishing conjugacy classes; 20E45 -- 20C15; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2020.69
• Languages: English
• ISSNs: 0308-2105

Abstract

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 .