Criterion of nonsolvability of a finite group and recognition of squares of simple groups

Zh. Wang, A. V. Vasil'ev, M. A. Grechkoseeva, A. Kh. Zhurtov

Published 2022-02-01Version 1

The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if among the prime divisors of the order of a group $G$, there are four different primes such that $\omega(G)$ contains all their pairwise products but not a product of any three of these numbers, then $G$ is nonsolvable. Applying this result we show that the direct square $Sz(q)\times Sz(q)$ of a simple exceptional Suzuki group $Sz(q)$ is uniquely characterized by its spectrum in the class of finite groups.