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Thread: unique subgroups

  1. #1
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    unique subgroups

    Prove that $\displaystyle A_4$ has a unique subgroup $\displaystyle V$ of order 4.

    I know all 12 elements of $\displaystyle A_4$, and I know $\displaystyle V$ contains $\displaystyle { 1, (12)(34), (13)(24), (14)(23)}$, but how do I prove that $\displaystyle V$ is a unique subgroup?
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  2. #2
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    Quote Originally Posted by dori1123
    Prove that $\displaystyle A_4$ has a unique subgroup $\displaystyle V$ of order 4.

    I know all 12 elements of $\displaystyle A_4$, and I know $\displaystyle V$ contains $\displaystyle { 1, (12)(34), (13)(24), (14)(23)}$, but how do I prove that $\displaystyle V$ is a unique subgroup?
    Since subgroups of order $\displaystyle 4$ are Sylow $\displaystyle 2$-subgroups it follows all subgroups of order $\displaystyle 4$ must be conjugate by Sylow's second theorem. Now $\displaystyle V$ is a normal subgroup of $\displaystyle A_4$. Therefore, it must be the only one.
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