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Math Help - Prove C is a subgroup of G...

  1. #1
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    Prove C is a subgroup of G...

    By the center of a group G we mean the set of all the elements of G which commute with every element of G, that is,

    C = {a e G : ax = xa for every x e G}

    Prove that C is a subgroup of G.

    Any advice? I have no idea how to do this...
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  2. #2
    MHF Contributor Swlabr's Avatar
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    When proving a given set, H, is a subgroup there are two things you need to prove,

    Given a, b \in H then,

    a.b \in H, and

    a^{-1} \in H.

    So, let g \in G and a, b \in Z(G), the center of G.

    Does g(ab) = (ab)g? Yes, it does, but show it.

    Does ga^{-1} = a^{-1}g? Yes, it does. However, there is a `trick' to this one. You need to remember that if A=B then A^{-1} = B^{-1}, and that ag^{-1} = g^{-1}a as a \in Z(G).
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