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    Stabilizers need not be normal

    P-TQGAM

    Area
    Algebra
    Topics
    Counterexamples, Normal Subgroups, Orbit-Stabilizer
    Status
    draft
    • Show that the stabilizer of an element need not be a normal subgroup?
    Solution

    Let \(S_3\) act naturally on \(\{1,2,3\}\). The stabilizer of \(1\) is \[ G_1=\{e,(23)\}. \] But \[ (12)(23)(12)^{-1}=(13)\notin G_1. \] Hence \(G_1\) is not invariant under conjugation, so \[ G_1\not\trianglelefteq S_3. \] Thus stabilizers need not be normal subgroups.

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