Order Of Abellian Group Non Abellian Group


1)Every group of order p^2 is abelian.

2)Suppose o(G)=pq, p<q and p does not divide q−1 then every group of order pq is cyclic and hence abelian (as every cyclic group is abelian).
Suppose p divides q−1 then there is a unique non-abelian group of order pq,obviously upto isomorphism.

3)For any prime there exist a nonabelian group of order p^3, namely Heisenberg Group with entries from a field of order p.


Can you suggest some questions from previous year asked on this topic of Abellian Group ?