I solved the 1st half of this question and I am stuck on the second part

Let be a group and let be the center of . For every , we'll define .

1. Show that for every , .

2. If ( a prime number), show that

I showed part 1 by induction on and by showing that the center of is isomorphic to therfore it is normal in .

For part 2, I'm thinking that since then all that is left to show is that , but I can't figure out a way to do it.

Appreciate any ideas

SK