1. The problem statement, all variables and given/known data If p is a prime and G is group of order p^2, then show that G is abelian. 2. Relevant equations n/a 3. The attempt at a solution I first consider Z(G), the centre of G. Since it is a normal subgroup of G, then by Lagrange's Theorem, |Z(G)| divides |G|. Hence |Z(G)| = 1, p or p^2. We know that Z(G) not the trivial subgroup (proof already given) hence it must be of order p^2 or p. If |Z(G)| = p^2, then Z(G) = G and hence by definition it is abelian. If |Z(G)| = p, then .... well this is where I am stuck! :( Please help!