1. The problem statement, all variables and given/known data (1)To prove this I have to let G be a group, with |G|=p^2. (2)Use the G/Z(G) theorem to show G must be Abelian. (3) Use the Fundamental Theorem of Finite Abelian Groups to find all the possible isomorphism types for G. 2. Relevant equations Z(G) = the center of G (a is an element of G such that ax=xa for all x in G) 3. The attempt at a solution I can prove it by using Conjugacy classes and gettting that the order of Z(G) must be non-trival and going on from there, but we have not gotten to Conjugacy Classes yet so i can't use this fact. Can anyone help me on this? I know that |Z(G)|=1 or pq when the order of |G|=pq where p and q are not distinct primes. From there I am unsure on how to uise the G/Z(G) thereom to prove that G is abelian.