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Math Help - Subgroup of S4

  1. #1
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    Subgroup of S4

    How can I show that S_4 has a subgroup isomorphic to D_8? Can someone help please?
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  2. #2
    Junior Member nimon's Avatar
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    I wonder who set you this question, because it would seem impossible - indeed it can be shown that no subgroup of S_{4} is isomorphic to D_{8}.

    Note that \#D_{8}=16, eight rotations and eight reflections, and that \#S_{4}=4!=24. By Lagrange's theorem, if G is a subgroup of S_{4}, then \#G|S_{4}, and it is very clear that isomorphic groups have the same number of elements. But 24 is not divisible by 16, so no subgroup of S_{4} is isomorphic to D_{8}.

    Maybe it was a trick question!
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  3. #3
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    Maybe he means D_4: some authors write D_{2n} for D_n because that shows the order of the group, while others find the notation D_n more appropriate because of its geometric meaning (and would argue that you write S_n and not S_{n!} !).

    In this case it is true, the 2-sylow subgroups of S_4 are isomorphic to D_4 (or D_8 if its order is 8).

    Think of the symmetries of a square whose vertices are nammed 1,2,3,4 , take a close look at the subgroup generated by \{(1234), (12)(34)\} for instance.
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