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Math Help - classify groups order 77

  1. #1
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    classify groups order 77

    This is a neat question [there is a fast solution]:

    Classify, up to isomorphism, groups of order 77.
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  2. #2
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    Quote Originally Posted by GaloisTheory1 View Post
    This is a neat question [there is a fast solution]:

    Classify, up to isomorphism, groups of order 77.
    My attempted solution is as follows:

    1. Z_{77} = Z_{7} \times Z_{11}.

    Since 11 is not congruent to 1modulo7, it has one Sylow 7-subgroup and one Sylow 11-subgroup, which has an intersection at {e}. Thus, the group of order 77 is the direct product of a normal Sylow 7-subgroup and a normal Sylow 11-subgroup, which is cyclic.

    I think there is only one group of order 77 up to isomorphism.
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  3. #3
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    Quote Originally Posted by GaloisTheory1 View Post
    This is a neat question [there is a fast solution]:

    Classify, up to isomorphism, groups of order 77.
    As aliceinwonderland said this is a special case of |G| = pq where p,q are odd primes.
    There some conditions on pq (what alice did) that force G\simeq \mathbb{Z}_{pq}.
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