A dual version of Huppert's conjecture on conjugacy class sizes

Abstract

In [J. Algebra 344 (2011), 205–228], a conjecture of J. G. Thompson for PSL

An Analogue of Huppert’s Conjecture for Character Codegrees

“…Analogues of Huppert's conjecture can be proposed for any set of integers related to a finite group. For instance, a dual version of Huppert's conjecture for the set of conjugacy class sizes is considered in [1][2][3] and verified for some families of simple groups such as PSL (2, q).…”

Equivalent version of Huppert's conjecture for $K_3$-groups

“…For instance, a dual version of Huppert's conjecture for the set of conjugacy class sizes is considered. For more results see [1,2,3]. In this paper, we are concerned with the following conjecture, inspired by Huppert's conjecture: Conjucture: Let G be a finite group and H a non-abelian simple group.…”

“…, where G ∼ = U 3 (3) or U 4 (2), we see that cod(G) contains a prime, a contradiction. Thus G is perfect and (2). Also by [10], we know that Mult(U 3 (3)) = 1 and Mult(U (2).…”

“…Thus G is perfect and (2). Also by [10], we know that Mult(U 3 (3)) = 1 and Mult(U (2). Also by [10], we know that cd(2.U 4 (2)) = {1, 4, 20, 36, 60, 64, 80} and we have a contradiction.…”

PGL2(q) cannot be determined by its cs

“…In 2015, it has been investigated that the above problem is true when S PSL 2 (q) [8]. Then, in [7], it has been proven that the answer of the above problem is true for many families of finite simple groups.…”