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Math Help - Subgroups of order 4 of S4

  1. #1
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    Subgroups of order 4 of S4

    I'm working on building a list of the normal subgroups of order 4 of S_4
    I know that all subgroups of order 4 are either isomorphic to \mathbb{Z}_4 or \mathbb{Z}_2 \oplus \mathbb{Z}_2

    I want to show that any subgroup of order 4 that is isomorphic to \mathbb{Z}_4 will not be normal.

    Since \mathbb{Z}_4 is cyclic, the subgroups in question must also be cyclic. So these groups must be generated by a permutation of S_4 of length 4.

    For example, consider H = <(1234)>
    I want to show that \exists g \in S_4 such that g(1234)g^{-1} \notin H.
    For ease of calculation I use g = (12).
    We see that H = \{ e, (1234), (13)(24), (1432) \}
    yet g(1234)g^{-1} = (134) \notin S_4
    So H is not normal

    I want to generalize this for any cyclic subgroup of S_4 but I cannot think of a way other than brute force to do this. I only want a hint on this, a push in the right direction.
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    Quote Originally Posted by Haven View Post
    I'm working on building a list of the normal subgroups of order 4 of S_4
    I know that all subgroups of order 4 are either isomorphic to \mathbb{Z}_4 or \mathbb{Z}_2 \oplus \mathbb{Z}_2

    I want to show that any subgroup of order 4 that is isomorphic to \mathbb{Z}_4 will not be normal.

    Since \mathbb{Z}_4 is cyclic, the subgroups in question must also be cyclic. So these groups must be generated by a permutation of S_4 of length 4.

    For example, consider H = <(1234)>
    I want to show that \exists g \in S_4 such that g(1234)g^{-1} \notin H.
    For ease of calculation I use g = (12).
    We see that H = \{ e, (1234), (13)(24), (1432) \}
    yet g(1234)g^{-1} = (134) \notin S_4
    So H is not normal


    Almost correct but not quite: g(1234)g^{-1}=(1342)\notin <(1234)> , so you still get that the subgroup isn't normal.
    Remember that conjugating ANY cycle by ANY permutation gives another cycle of EXACTLY the same length as the original one, and also that any two cycles of the same length in S_n are conjugated iff they've the same length. This also answers your question below.

    Tonio

    I want to generalize this for any cyclic subgroup of S_4 but I cannot think of a way other than brute force to do this. I only want a hint on this, a push in the right direction.
    .
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