Hi, I have two ways to prove symmetric group is not cyclic if n is more than 2. I'd like to listen to your ideas about my proofs. Thanks. 1) Since symmetric group is not abelian (n>2), and cyclic group is abelian, so, symmetric group is not cyclic group (n>2) 2) Since symmetric group is a permutation group of order n!, and cyclic group only has order n, so, when n>2, symmetric group is not cyclic. Please give your suggestion. Thanks again. P.L.