On characterization of a finite group by the set of conjugacy class sizes

Let [Formula: see text] be a finite group and [Formula: see text] be the set of its conjugacy class sizes. In the 1980s, Thompson conjectured that the equality [Formula: see text], where [Formula: see text] and [Formula: see text] is simple, implies the isomorphism [Formula: see text]. In a series of papers of different authors, Thompson’s conjecture was proved. In this paper, we show that in some cases it is possible to omit the conditions [Formula: see text] and [Formula: see text] is simple and prove a more general result.

Finite groups with 4p 2 q elements of maximal order

It is an interesting and difficult topic to determine the structure of a finite group by the number of elements of maximal order. This topic is related to Thompson’s conjecture, that is, if two finite groups have the same order type and one of them is solvable, then the other is solvable. In this article, we continue this work and prove that if G G is a finite group which has 4 p 2 q 4{p}^{2}q elements of maximal order, where p p , q q are primes and 7 ≤ p ≤ q 7\le p\le q , then either G G is solvable or G G has a section who is isomorphic to one of L 2 ( 7 ) {L}_{2}\left(7) , L 2 ( 8 ) {L}_{2}\left(8) or U 3 ( 3 ) {U}_{3}\left(3) .

Thompson’s conjecture on conjugacy class sizes for the simple group PSUn(q)

We show that if [Formula: see text] is a finite centerless group with the same conjugacy class sizes as [Formula: see text], then [Formula: see text] and so verify a conjecture attributed to John G. Thompson.

On Thompson’s conjecture for alternating groups of large degree

For a finite group