 Burnside's "pa qb" theorem states that if the order of a group is the product of one or two prime powers, then it is solvable, and so the group is not simple, or is of prime order and is cyclic.This rules out every group up to order 30 .If "G" is simple, and |"G"| = 30, then "n"3 must divide 10 ( = 2 · 5), and "n"3 must equal 1 (mod 3).