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Thread: Show a subgroup is normal

  1. #1
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    Show a subgroup is normal

    Hey everyone,

    I have an exercise in which I need to show the subgroup H \subset S_4 is normal, when H is given as:

    H = \lbrace e, (12)(34),(13)(24),(14)(23) \rbrace

    I've maneged to show H is indeed a subgroup and my idea is to somehow use the conjugate of a subgroup, but can't seem to make it work. Is there a better way to tackle this problem?

    Thanks in advance.
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  2. #2
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    Re: Show a subgroup is normal

    Quote Originally Posted by Krisly View Post
    I have an exercise in which I need to show the subgroup H \subset S_4 is normal, when H is given as:
    H = \lbrace e, (12)(34),(13)(24),(14)(23) \rbrace
    I've maneged to show H is indeed a subgroup and my idea is to somehow use the conjugate of a subgroup, but can't seem to make it work. Is there a better way to tackle this problem?
    I don't know of any quick solution for this task.
    There is a theorem from which you may be able to adapt proof:
    A subgroup $\mathscr{S}$ of a group $\mathscr{G}$ is normal iff all of its right-cosets are left-cosets.
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  3. #3
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    Re: Show a subgroup is normal

    A fact which is not hard to prove: in a symmetric group, a conjugate of a cycle is another cycle of the same length. So for example $g(1\,2)(3\,4)g^{-1}=g(1\,2)g^{-1}g(3\,4)g^{-1}$, another product of two 2 cycles. Now notice in $S_4$, the only products of two 2 cycles are the non-identity elements of H.
    Thanks from topsquark, Plato and Krisly
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    Re: Show a subgroup is normal

    Thank you very much! does that mean I can argue the following using both comments?

    The group H is normal iff the left cosets equals the right cosets hence;

    f H = H f, \: \forall f \in S_4, thus H is normal iff fhf^{-1} \in H \: \forall f \in S_n \land h \in H .


    And as  f(ab)(cd)f^{-1} = f (ab) f^{-1} f (cd) f^{-1} = (f[a]f[b])(f[c]f[d])

    in addition to the argument all composition of disjoint two cycles of S_4 belongs to the set H, then fHf^{-1} \in H, thus H is normal?
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  5. #5
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    Re: Show a subgroup is normal

    Yes. As Plato points out a subgroup H of a group G is normal iff for all g in G, gH=Hg.
    Now for any g in G, gH=Hg iff $gHg^{-1}=H$ iff for any g in G and any h in H
    $ghg^{-1}\in H$.
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