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

The spectrum ?(G) of a finite group G is the set orders its elements. following sufficient criterion nonsolvability proved: if, among prime divisors order G, there are four different primes such that contains all their pairwise products but not product any three these numbers, then nonsolvable. Using this result, we show for q ? 8 and ? 32, direct square Sz(q) × simple exceptional Suzuki uniquely characterized by in class groups, while Sz(32) Sz(32), exactly groups with same spectrum.