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Math Help - groups

  1. #1
    Junior Member
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    groups

    Classify groups of order 6 by analyzing the following three cases.
    (a) G contains an element of order 6
    (b) G contains an element of order 3 but none of order 6
    (c) All elements of G have order 1 or 2


    could anyone please help me with this problem ?
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  2. #2
    Super Member Gamma's Avatar
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    a) if an element has order 6, in a group of order 6, it generates the whole group, i.e. this is \mathbb{Z}_6
    b) This would be the lone nonabelian group of order 6. S_3 or D_6 they are the same.
    c) I don't think there is such a group, the sylow 3 subgroup is normal and they have trivial intersection so HK is a subgroup of order 6, so G must be a semi direct product, ie one of the two i mentioned before. Alternatively, cauchy's theorem guarantees the existence of an element of order 3.
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