Let be a group. Prove that if for some primes and , then either is abelian (i.e. ) or .
Suppose . Since , by Lagrange’s Theorem, divides . Thus is one of , , or . If , then since is finite, is abelian. If (without loss of generality) , then by Lagrange’s Theorem , so that is cyclic. By a previous theorem, then, is abelian.