A Criterion for Nonsolvability of a Finite Group and Recognition of Direct Squares of Simple Groups

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 not equal 32, the direct square Sz(q) x 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) x Sz(32), there are exactly four finite groups with the same spectrum.