Proving Group of Order p^2 is Cyclic or ZpXZp

[halvizo1031]

Homework Statement

If the order of G is p^2 and p is prime, then show that G is either cyclic or isomorphic to ZpXZp...

Homework Equations

The Attempt at a Solution

Any hints here will help!

 

[VeeEight]

I think two helpful facts here is that if the |G|=p² for prime p, then G has non-trivial Centre. Furthermore, the Normalizer of the group is greater than the Centre. You can use this to show that the Centre of G is the entire group G, which implies it is abelian. Then use some other theorems involving abelian groups to prove your theorem.