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Math Help - Prove C(a) is a subgroup of G

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

    Let G be a group and a an element of G. the centralizer of a is the set C(a)={g in G: ga=ag}. prove that C(a) is a subgroup of G.
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  2. #2
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    Quote Originally Posted by nikie1o2 View Post
    Let G be a group and a an element of G. the centralizer of a is the set C(a)={g in G: ga=ag}. prove that C(a) is a subgroup of G.

    What've you tried? Prove that the group's unit is contained in C(a) and also that a,b\in C(a)\Longrightarrow ab^{-1}\in C(a)

    Tonio
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  3. #3
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    Quote Originally Posted by nikie1o2 View Post
    Let G be a group and a an element of G. the centralizer of a is the set C(a)={g in G: ga=ag}. prove that C(a) is a subgroup of G.
    Let x,y\in C(a). Then x=axa^{-1} and y=aya^{-1}. So xy=axa^{-1}aya^{-1}=axya^{-1}, and xy\in C(a). Now observe that (ax^{-1}a^{-1})x=(ax^{-1}a^{-1})(axa^{-1})=e. So x^{-1}=ax^{-1}a^{-1}, that is, x^{-1}\in C(a). Since clearly e\in C(a), then it follows that C(a) is a subgroup of G
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