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Math Help - Simple question about a definition

  1. #1
    Senior Member slevvio's Avatar
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    Simple question about a definition

    I have this in my lecture notes:

    Let G be a group and let P and Q be p-subgroups. Suppose that Q normalizes
    P. Then PQ is also a p-subgroup.

    However it doesnt say anywhere what normalizes means, I know what the normaliser is

    N_G (Q) = \{ g \in G : gQ = Qg \}, N_G (P) = \{ g \in G : gP = Pg \}.

    I was wondering if anyone could tell me what Q normalises P means? Thanks very much
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  2. #2
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    Quote Originally Posted by slevvio View Post
    I have this in my lecture notes:

    Let G be a group and let P and Q be p-subgroups. Suppose that Q normalizes
    P. Then PQ is also a p-subgroup.

    However it doesnt say anywhere what normalizes means, I know what the normaliser is

    N_G (Q) = \{ g \in G : gQ = Qg \}, N_G (P) = \{ g \in G : gP = Pg \}.

    I was wondering if anyone could tell me what Q normalises P means? Thanks very much

    Q normalizes P\iff Q\subset N_G(P)\iff P^q=q^{-1}Pq=P\,\,\,,\,\forall\,q\in Q

    Tonio
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