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Math Help - conjugate, central element and Normal subgroup

  1. #1
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    conjugate, central element and Normal subgroup

    Can you help me to prove this,

    Let G be a finite Group, and (a G). Prove that if a has two conjugates in G, then Z(a) normal subgroup of G.

    ( Z(a) is the central of a )
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by alper82 View Post
    Can you help me to prove this,

    Let G be a finite Group, and (a G). Prove that if a has two conjugates in G, then Z(a) normal subgroup of G.

    ( Z(a) is the central of a )
    Have you come across the orbit-stabiliser theorem? If so, then use this, and remember that a subgroup of index 2 is normal...
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