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Math Help - Centers

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    Centers

    Must the center of a group be made up of congugacy classes?
    Last edited by alexandrabel90; May 13th 2011 at 01:20 PM.
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    Quote Originally Posted by alexandrabel90 View Post
    Must the center of a group be made up of congugacy classes?
    What does this mean? Is it true that the center of a group is a union of conjugacy classes? The answer is yes for two reasons. The first reason is the fact that \mathcal{Z}(G)\unlhd G for any group G (since \mathcal{Z}(G)=\ker\phi where \phi:G\to \text{Aut}(G):g\mapsto i_g where i_g is the inner automorphism) and normal subgroups are subgroups which are unions of conjugacy classes. Perhaps simpler is the fact that, by definition, \displaystyle \mathcal{Z}(G)=\bigsqcup_{C_g: \text{card}(C_g)=1}C_g where C_g is the conjugacy class of g and the square union is just a fancy way of saying that the sets in the union are pairwise disjoint.
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