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Math Help - Prove that for any a E G, Ca is a subgroup of G.

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    Prove that for any a E G, Ca is a subgroup of G.

    For a fixed element a of a group G, the set Ca = {x E G | ax = xa } is the centralizer of a in G.

    Prove that for any a E G, Ca is a subgroup of G.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by rainyice View Post
    For a fixed element a of a group G, the set Ca = {x E G | ax = xa } is the centralizer of a in G.

    Prove that for any a E G, Ca is a subgroup of G.
    Where is it you are stuck on this question?

    Let x, y \in C_a. Then you need to check that xy \in Ca and x^{-1} \in Ca. This is precisely what you would always do to check something is a subgroup.

    Checking that xy is in C_a is almost elementary (substitute `xy' in the above condition for `x), but the fact that x^{-1} \in C_a is not immediately obvious. However, just take the condition, and pre- and post-multiply by x^{-1} to get that the condition also holds for x^{-1}
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    Quote Originally Posted by Swlabr View Post
    Where is it you are stuck on this question?

    Let x, y \in C_a. Then you need to check that xy \in Ca and x^{-1} \in Ca. This is precisely what you would always do to check something is a subgroup.

    Checking that xy is in C_a is almost elementary (substitute `xy' in the above condition for `x), but the fact that x^{-1} \in C_a is not immediately obvious. However, just take the condition, and pre- and post-multiply by x^{-1} to get that the condition also holds for x^{-1}

    Currently, this is my first time start to know Group Theory. I don't really know how to prove if it is a subgroup or not because I do not know where and how I should start...
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    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by rainyice View Post
    Currently, this is my first time start to know Group Theory. I don't really know how to prove if it is a subgroup or not because I do not know where and how I should start...
    It'll be in your notes. To prove that H is a sugroup of G you have to prove two things,

    That for all g, h \in H then g*h \in H, and for all h \in H, h^{-1} \in H.
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    Quote Originally Posted by Swlabr View Post
    It'll be in your notes. To prove that H is a sugroup of G you have to prove two things,

    That for all g, h \in H then g*h \in H, and for all h \in H, h^{-1} \in H.
    And you'll also need to show there's at least something in the group, usually e\in H suffices. The reason is that the empty set satisfies those two conditions but it's not a group
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