A proof of Szep's conjecture on nonsimplicity of certain finite groups (Q1820236)

The following confirms a conjecture of J. Szep. Theorem. If the finite group \(G=AB\) is the product of two subgroups A and B with nontrivial centers, then G is not nonabelian and simple. As a corollary it follows that the cardinalities of any two nontrivial conjugacy classes in a finite nonabelian simple group cannot be relatively prime. The proof of these results uses heavily the classification of finite simple groups.