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Properties of the internal direct product of subgroups

  1. Mar 30, 2010 #1
    1. The problem statement, all variables and given/known data
    Let H and K be groups and let G = H x K. Recall that both H and K appear as subgroups of G in a natural way. Show that these subgroups H (actually H x {e}) and K (actually {e} x K) have the following properties:
    a: every element of G is of the form hk for some h in H and k in K.
    b: hk = kh for all h in H and k in K.
    c: H INTERSECT K = {e}.


    2. Relevant equations



    3. The attempt at a solution
    first off, i'm not sure what "Recall that both H and K appear as subgroups of G in a natural way" means. could someone explain that?
    now, a seems easy enough. dealing with direct product of H and K defines elements of G as hk for some h and k in H and K respectively.
    c, if i'm understanding it right, should actually be like {e} x {e}, since h looks like (h,e) and k looks like (e,k). these are both subgroups, so they both have e, and they would only coincide at {e} x {e}, or (e,e).
    b, i have no idea. i'm not told G is abelian. the problem i did before this was to show that IF G is abelian, then the direct product is abelian, but... any clues?
     
  2. jcsd
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