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Math Help - Conjugates in a finite group

  1. #1
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    Conjugates in a finite group

    Show that in a finite group of order n, an element of order k has at most n/k conjugates.

    Proof so far.

    Let G be a group with order n, and suppose that g is an element with the order of k. Then I have k|n by the property of order. But I'm not too familiar in dealing with conjugates, how should I get started here? Thanks.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Show that in a finite group of order n, an element of order k has at most n/k conjugates.
    Let G act on itself by conjugative. Then the conjugates to a\in G is the orbit Ga. But |Ga| = |G:G_a|. And |G_a| = |C(a)| - the centralizer. But \{ a,a^2,...,a^k\} \subseteq C(a). Therefore, |C(a)| \geq k. This means [G:G_a] \leq n/k. Therefore, |Ga| \leq n/k.
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