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Classifying finite groups

  1. Nov 8, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that the group of order 175 is abelian.


    2. Relevant equations



    3. The attempt at a solution
    |G|=175=527. Using the Sylow theorems it can be determined that G has only one Sylow 2-subgroup of order 25 called it H and only one Sylow 7-subgroups called it K. Thus, H and K are normal subgroups of G and G=H x K which is isomorphic to the direct product of H and K. Since |H|=52, then H is Abelian. Since K is of prime order then K is cyclic and therefore also Abelian.

    I am not sure whether I can now conclude that G must be abelian since it is the external (or direct) product of abelian subgroups.
     
  2. jcsd
  3. Nov 8, 2009 #2
    Can you try to prove that the direct product of abelian groups is abelian?
     
  4. Jan 7, 2010 #3
    direct products of abelian groups are abelian, this is obvious. Look at commutators and use the fact that they have trivial intersection.
     
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