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Math Help - Check center of group proof

  1. #1
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    Check center of group proof

    Question-Let G be a group and let g∈G. If z ∈ Z(G) (the center of a group G), show that the inner automorphism induced by g is the same as the inner automorphism induced by zg(that is ϕg = ϕzg) Hint: What is the definition of Z(G)?

    Proof:
    Since Z(G) just contains e, then z must equal e
    so ϕg = ϕzg
    ϕg=ϕeg
    ϕg=ϕg

    I'm sure I'm missing some stuff, but is this even close?
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  2. #2
    Member ModusPonens's Avatar
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    Re: Check center of group proof

    Hint: z commutes with all elements of G.
    Thanks from Johngalt13
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  3. #3
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    Re: Check center of group proof

    no, Z(G) can be lots bigger than just {e}. in fact, if G is abelian Z(G) is all of G!

    however, for any z in Z(G), we have zx = xz for any element x of G. and this means that:

    \phi_z(x) = zxz^{-1} = xzz^{-1} = xe = x for any x in G, that is, \phi_z is the identity automorphism of G.

    you'll be done if you can prove that \phi_g \circ \phi_h = \phi_{gh} for any g,h in G (why?)
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