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Almost commutative

  1. Jun 30, 2012 #1
    1. The problem statement, all variables and given/known data
    I'm trying to figure out if the following property has a name:

    for [itex]g\in G[/itex], [itex]h\in H[/itex], [itex]\exists h'\in H[/itex] s.t. gh=h'g.

    obviously this is not quite commutativity, but it seems like it might be useful in a variety of situations.

    2. Relevant equations

    I've just finished a proof that if a group K has two normal subgroups G and H, whose intersection is just the identity, and whose join is K, then there exists an isomorphism θ(g,h)=gh for all g in G and all h in H. The key to proving surjectivity involved the fact that since H is normal, ghg[itex]^{-1}[/itex] is also in H (call this h') so h=g[itex]^{-1}[/itex]h'g and

    [itex]gh=gg^{-1}h'g=h'g[/itex]

    3. The attempt at a solution
    I think I've seen this discussed elsewhere, just can't remember the name. ----commutativity?
     
  2. jcsd
  3. Jun 30, 2012 #2

    HallsofIvy

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    For H a group, the "property" that there exists such an h' doesn't have a name because it is always true! For any such g and h, [itex]h'= ghg^{-1}[/itex] must exist. And that is saying that h and h' are conjugates. Perhaps that is what you are looking for.

     
  4. Jun 30, 2012 #3
    I understand that in the context of groups, this h' is just the result of conjugation of h by g, but my thought was that perhaps this might occur outside of the context groups (say in cases where inverses may not exist)? I can't think of any examples of this though... maybe this is just not useful.
     
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