Finite groups with 4p 2 q elements of maximal order

Abstract 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) .