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

Abstract

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. Using this result, we show that for $q⩾8$ and $q\ne 32$, the direct square $Sz(q)\times Sz(q)$ of the simple exceptional Suzuki group $Sz(q)$ is uniquely characterized by its spectrum in the class of finite groups, while for $Sz(32)\times Sz(32)$, there are exactly four finite groups with the same spectrum.

In the third version, Theorem 1 is slightly reformulated and some references are corrected

References in corpus (2)

• On finite groups isospectral to simple classical groups
• Finite groups isospectral to simple groups

Cited by in corpus (2)

• On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups
• On recognition of direct powers of finite simple linear groups by spectrum