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Math Help - Subgroups, isomorphic

  1. #1
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    Subgroups, isomorphic

    Let H be a subgroup of G and let a be in G. Show that aHa^-1 is a subgroup of G that is isomorphic to H.
    I want to show aHa^-1 is 1-1, onto and has a homomorphism property.
    Ok so my problem with this is that with isomorphisms before I generally had a function that I worked from, so it was easier to show 1-1, onto, and c(ab)=c(a)c(b), but without a funcction, I get stuck beginning.
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  2. #2
    Super Member Gamma's Avatar
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    Isomorphism

    Let \phi:H\longrightarrow aHa^{-1} by \phi(x)=axa^{-1}
    Clearly onto and well defined.
    To check injectivity, suppose ax_1a^{-1}=ax_2a^{-1} then simply multiply on the left by a and on the right by a^{-1} to conclude x_1=x_2
    we need only check it is a homomorphism

    \phi(x)\phi(y)=axa^{-1}aya^{-1}=axya^{-1}=\phi(xy) proving the isomorphism
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