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Thread: 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, $\displaystyle H$, is a subgroup there are two things you need to prove,

    Given $\displaystyle a, b \in H$ then,

    $\displaystyle a.b \in H$, and

    $\displaystyle a^{-1} \in H$.

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

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

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