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Math Help - Direct product and Decomposition of a symmetric group

  1. #1
    Senior Member
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    Direct product and Decomposition of a symmetric group

    I am looking for a mechanism to find a decomposition of symmetric groups. For finitely generated abelian group G, there is a mechanism to decompose G such that G is isomorphic to a direct sum of cyclic groups.
    For symmetric groups, it seems a bit complex for me to find it.
    For example,

    (1) is it possible for S_{3}, S_{4}, S_{5} to be decomposed as a direct product of groups?
    In more general cases, is there any mechanism to decompose S_{n} as a direct product of groups?

    (2) is there any mechanism to find an isomorphic group of a direct product of symmetric groups? Let's say,
    G = S_{3} \times S_{4} \times S_{5}
    Any isomorphic group of above G?
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  2. #2
    Super Member Gamma's Avatar
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    Dec 2008
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    Iowa City, IA
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    I can give you a cheesy isomophism

    recall the dihedral group of order 6, D_6 \cong S_3 in fact every non abelian group of order 6 is isomorphic. So D_6 \times S_4 \times S_5 \cong S_3 \times S_4 \times S_5
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