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**kleenex** Let G be p-group and G acts on G by conjugation, show $\displaystyle Z(G) \neq <1>$.

I saw there is a lemma on a book say:

Let G be finite p-group (ie $\displaystyle |G|=p^n$) which acts on finite set $\displaystyle X$.

Let $\displaystyle F=${$\displaystyle x \in X | x^g =x \forall g \in G$} $\displaystyle =$ set of fixed points of $\displaystyle G$.

Then $\displaystyle |F| \equiv |X| $(mod p).

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