Let the order of G be pq where p and q are primes (not necessarily distinct). Show that the center has order 1 or pq. Lets call the center of G Z(G)
By lagrange we know the order of Z(G) to be 1, p, q or pq. Now I must show it cannot be p or q.
I found a nifty couple of theorems in my algebra book that says: If G/Z(G) is cyclic then G is abelian. Using the contrapositive and lagrange we can show that a non abelian group of order pq must have a trivial center.
I don't know how to prove that second statement but it seems like that would be half of my proof. Then if I can show that an Abelian group with order pq must have a center of order pq i would be done.
Can you guys help, give some hints, or perhaps other ways you would do this? Thanks!!