On solvable groups with one vanishing class size

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.

On finite groups with exactly one vanishing conjugacy class size

“…Consequently, n 41 , n 42 = 1. As was mentioned above, 2 and θ 4 (n 42 ) = θ 4 (n 41 ) 2 . Moreover, suppose that θ 4 (n 41 ) = θ 3 (n 31 ).…”

“…Otherwise, m = 0. Note that Z n (a) = {1}, unless (a, n) ∈ { (2,1), (2,6), (−2, 2), (−2, 3), (3,1), (−3, 2)}, by [8].…”

“…In [2], Bianchi, Camina, Lewis and Pacifici classify the finite super-solvable groups with one vanishing conjugacy class size and put forward a problem on the solvability of the groups with one vanishing conjugacy class size. In this paper, we prove that:…”