group/ subgroups?

Question

Let G be a group of order p^n , where p is prime and n is a positive integer. Show that G contains a normal subgroup of order p^k for each k with 0 <= k <= n. The hint is to use the fact that Z(G) (the center of G) is nontrivial and induction.

Answer 1

I believe that this proof uses the class equation to show that the center is nontrivial. You are induction either on the order of the center or on the order of G, I am not sure. Maybe you look for a min counter ex.

I believe that this problem is worked out in Dummit and Foote.

Answer 2

add it..