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    $[A_4,A_4]\cong\ZZ_2^2$

    P-XVHAO

    Area
    Algebra
    Topics
    Commutators, Permutations
    Status
    draft

    Show that \[ [A_4,A_4]\cong C_2\times C_2. \]

    Solution

    Let \[ V_4=\{1,(12)(34),(13)(24),(14)(23)\}\trianglelefteq A_4. \] Since \[ A_4/V_4\cong C_3 \] is abelian, the commutator subgroup satisfies \[ [A_4,A_4]\le V_4. \]

    It remains to show that the commutator subgroup is nontrivial. Let \[ a=(123),\qquad b=(124). \] Then \[ [a,b]=aba^{-1}b^{-1}=(12)(34). \] Thus \[ (12)(34)\in[A_4,A_4]. \] The commutator subgroup is normal in \(A_4\), and the three nonidentity elements of \(V_4\) are conjugate in \(A_4\). Hence all three double transpositions lie in \([A_4,A_4]\).

    Therefore \[ [A_4,A_4]=V_4\cong C_2\times C_2. \]

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