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Math Help - Centralizer proof

  1. #1
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    Centralizer proof

    Show <a> is a subset of C(a) where C(a) is such that xa=ax.


    Knowing xa=ax tells me we have an abelian group.
    <a> is a cyclic subgroup
    we have x=a^n . Need to show x=a^n is a subset of xa=ax. Not sure how to do that
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  2. #2
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    I don't know, but this is really frustrating me.
    so x=a^n
    or x=a*a*a*a*a*a.........
    Since <a> is a subgroup, inverses exist
    Thus x=a^-1aaaaaaa.....
    x=a^n-1
    We eventually get x=1
    I'm not sure where I'm going wrong in my thought process
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  3. #3
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    Quote Originally Posted by kathrynmath View Post
    I don't know, but this is really frustrating me.
    so x=a^n
    or x=a*a*a*a*a*a.........
    Since <a> is a subgroup, inverses exist
    Thus x=a^-1aaaaaaa.....
    x=a^n-1
    We eventually get x=1
    I'm not sure where I'm going wrong in my thought process

    <a>\subset C_G(a)\Longrightarrow a^k\cdot a=a\cdot a^k\,,\,\forall k ...but this is obvious, ain't it?

    Tonio
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