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

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

    If H is a subgroup of G then the centralizer C(H) of H we mean the set C(H)={x is a member of G/ xh=hx for all h as a member of H}. Prove that C(H) is a subgroup of G.
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  2. #2
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    Quote Originally Posted by mandy123 View Post
    If H is a subgroup of G then the centralizer C(H) of H we mean the set C(H)={x is a member of G/ xh=hx for all h as a member of H}. Prove that C(H) is a subgroup of G.
    You know the definition of what it means being a subgroup? Just check all the conditions which need to be satisfied. This should be a straightforward problem, just post what you did.
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  3. #3
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    Question

    so I think i got it, but I am not sure!

    Let x be a member of G
    b=x and a=x
    ab^-1=xx^-1 which is a member of G, so e is a member of G

    If a=e then b is a member of G
    eb^-1 is a member of G so b^-1 is a member of H

    x,y are members of G
    x=a and y=b^-1
    b=y^-1
    b is a member of G so ab^-1=xy which is a member of G

    Will this work or am i missing some steps?
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  4. #4
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    Quote Originally Posted by mandy123 View Post
    Will this work or am i missing some steps?
    I am not really sure what you are doing. Here, watch this, look at how we prove closure property. Let a,b\in C(H) then it means ax = xa and bx=xb for all x\in H. Therefore, (ab)x = a(bx) = a(xb) = (ax)b = (xa)b = x(ab). Thus, ab\in C(H). And so it is closed.
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