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Math Help - p-group help

  1. #1
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    p-group help

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

    I saw there is a lemma on a book say:
    Let G be finite p-group (ie |G|=p^n) which acts on finite set X.
    Let F={ x \in X | x^g =x \forall g \in G} = set of fixed points of G.
    Then |F| \equiv |X| (mod p).

    I think this lemma may help to solve this problem, but I need some help for that. Thank you
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  2. #2
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    Quote Originally Posted by kleenex View Post
    Let G be p-group and G acts on G by conjugation, show Z(G) \neq <1>.

    I saw there is a lemma on a book say:
    Let G be finite p-group (ie |G|=p^n) which acts on finite set X.
    Let F={ x \in X | x^g =x \forall g \in G} = set of fixed points of G.
    Then |F| \equiv |X| (mod p).

    I think this lemma may help to solve this problem, but I need some help for that. Thank you
    Let G act upon itself (call this set X) by conjugation. Since G and X are finite and G is a p-group it means |G| \equiv |X^G| (\bmod p) where X^G is the invariant subset fixed by G. In fact, X^G is precisely Z(G) by how we defined the group action. This tells us that |G| \equiv |Z(G)| (\bmod p) since LHS is divisible by p it means p divides Z(G).
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