Finite groups with 4p2q 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 GG is a finite group which has 4p2q4{p}^{2}q elements of maximal order, where pp, qq are primes and 7≤p≤q7\le p\le q, then either GG is solvable or GG has a section who is isomorphic to one of L2(7){L}_{2}\left(7), L2(8){L}_{2}\left(8) or U3(3){U}_{3}\left(3).

Keywords

• finite groups
• solvable groups
• the order of elements
• 05c25