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Math Help - Inner Automorphisms

  1. #1
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    Inner Automorphisms

    I'm just having an off night and while reading through an algebra book I came across that center Z(G) is the kernel in this, but I can't verify it to myself. Can anyone verify the reason for this?

    This is what I'm talking about: Center (group theory) - Wikipedia, the free encyclopedia
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  2. #2
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    Never mind. I figured it out. It follows since the center is a subgroup and has the property that x ∈Z(G), g∈G then gxg-1∈Z(G).
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  3. #3
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    Quote Originally Posted by bweiland View Post
    I'm just having an off night and while reading through an algebra book I came across that center Z(G) is the kernel in this, but I can't verify it to myself. Can anyone verify the reason for this?

    This is what I'm talking about: Center (group theory) - Wikipedia, the free encyclopedia
    You already figured that out? Anyhow, I don't want to discard this reply.

    Every element g of a group G, an inner automorphism \phi_g:G \rightarrow G defined by \phi_g(h)=ghg^{-1}. Verfiy that \phi_g is an automorphism of G for given h \in G. Now, let f:G \rightarrow \text{Aut(G)} defined by f(g)=\phi_g.The kernel of this map, which maps to an identity automorphism, is \{g \in G | \phi_g(h)=h \text{ for all } h \in G\}=\{g \in G | gh=hg \text{ for all } h \in G\}.
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  4. #4
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    I'm glad you still posted that reply because it reminded me that I've been writing my functions in more of an analysis format, which is just nonsense in algebra.
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bweiland View Post
    I'm glad you still posted that reply because it reminded me that I've been writing my functions in more of an analysis format, which is just nonsense in algebra.
    What's the analysis form of functions?
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  6. #6
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    I meant that I was writing my functions more like you would in calculus.

    So, for a mapping, I should have θ(g)=ιg(x) where ιg(x) = gxg-1 but instead I was writing θ(g) =gxg-1 and that makes no sense since it doesn't define x.
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  7. #7
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bweiland View Post
    I meant that I was writing my functions more like you would in calculus.

    So, for a mapping, I should have θ(g)=ιg(x) where ιg(x) = gxg-1 but instead I was writing θ(g) =gxg-1 and that makes no sense since it doesn't define x.
    I would really say that \theta_g(x)=gxg^{-1}. The subscript notation denotes that g is fixed upon inspection.

    P.S. That is how you write a lot of things in alysis
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  8. #8
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    Exactly, but in algebra it is poor notation.
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