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    $H$ characteristic in $K\trianglelefteq G$ implies $H\trianglelefteq G$

    E-AMD-JRUQKWKO

    Area
    Algebra
    Topics
    Automorphisms, Normal Subgroups, Subgroups
    Status
    draft

    Show that if \(H \leq K \trianglelefteq G\) and \(H\) is characteristic in \(K\), then \(H \trianglelefteq G\).

    Solution

    Let \(g\in G\). Since \(K\trianglelefteq G\), conjugation by \(g\) restricts to an automorphism \[ c_g:K\longrightarrow K, \qquad k\longmapsto gkg^{-1}. \] Because \(H\) is characteristic in \(K\), every automorphism of \(K\) preserves \(H\). Hence \[ gHg^{-1}=c_g(H)=H. \] This holds for every \(g\in G\), so \[ H\trianglelefteq G. \]

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