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