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    $A_n$ is normal in $S_n$

    P-BOSWW

    Area
    Algebra
    Topics
    Normal Subgroups, Permutations
    Status
    draft
    • Prove that \(A_n\) is normal in \(S_n\).
    Solution

    Consider the sign homomorphism \[ \operatorname{sgn}:S_n\longrightarrow\{\pm1\}. \] Its kernel is exactly \(A_n\). Kernels of group homomorphisms are normal, hence \[ A_n\trianglelefteq S_n. \] Equivalently, \(A_n\) has index \(2\) in \(S_n\), and every subgroup of index \(2\) is normal.

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