On Thompson Problem

TL;DR

This paper addresses Thompson's problem on determining the solvability of finite groups, providing a positive answer when the prime graph of the group is disconnected, based on group order and element orders.

Contribution

It offers a solution to Thompson's problem for groups with disconnected prime graphs, advancing the understanding of group solvability criteria.

Findings

Positive solution for Thompson's problem when prime graph is disconnected

Characterization of finite simple groups using order and element orders

Extension of group solvability criteria

Abstract

In 1987, the second author of this paper reported his conjecture, all finite simple groups $S$ can be characterized uniformly using the order of $S$ and the set of element orders in $S$, to Prof. J. G. Thompson. In their communications, Thompson posed his problem about the judgment of solvability of finite groups $G$. In this paper we give a positive answer for Thompson's problem if the prime graph of $G$ is not connection.