I need to prove that a group of order 15 is abelian. I have (tried to) attached my work but it's too long. Basically, I'm looking at the class equation and considering all possibile orders of the center Z(G). I've successfully eliminated the cases where |Z|=3,5. I'm having trouble with coming up with a contradiction when assuming |Z(G)|=1. I'm also interested in seeing alternative proofs (perhaps something more elegant), and also a proof that G is in fact cyclic. Is there any decent generalization of this problem?