Thread: p-group help

Let G be p-group and G acts on G by conjugation, show .

I saw there is a lemma on a book say:
Let G be finite p-group (ie ) which acts on finite set .
Let { } set of fixed points of .
Then (mod p).

I think this lemma may help to solve this problem, but I need some help for that. Thank you

Originally Posted by kleenex

Let G be p-group and G acts on G by conjugation, show .

I saw there is a lemma on a book say:
Let G be finite p-group (ie ) which acts on finite set .
Let { } set of fixed points of .
Then (mod p).

I think this lemma may help to solve this problem, but I need some help for that. Thank you

Let act upon itself (call this set ) by conjugation. Since and are finite and is a p-group it means where is the invariant subset fixed by . In fact, is precisely by how we defined the group action. This tells us that since LHS is divisible by it means divides .