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    $\Inn(G)$ is normal in $\Aut(G)$

    E-AMD-3LUW3P7R

    Area
    Algebra
    Topics
    Automorphisms, Normal Subgroups
    Status
    draft

    Show that \(\Inn(G) \normal \Aut(G)\).

    Solution

    For \(g\in G\), write \(c_g(x)=gxg^{-1}\). Then \(\Inn(G)=\{c_g:g\in G\}\).

    Let \(\phi\in\Aut(G)\). For every \(g,x\in G\), \[ (\phi c_g\phi^{-1})(x) =\phi\bigl(g\phi^{-1}(x)g^{-1}\bigr) =\phi(g)x\phi(g)^{-1} =c_{\phi(g)}(x). \] Hence \[ \phi c_g\phi^{-1}=c_{\phi(g)}\in\Inn(G). \] Therefore \(\Inn(G)\trianglelefteq\Aut(G)\).

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