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Distinct left(right) cosets

  1. Feb 16, 2010 #1
    G group, H subgroup of G.
    Suppose aH and bH are distinct leftcosets then Ha and Hb must be distinct right cosets?


    My humble thoughts:
    the left coset aH consists of a times everything in H;
    Ha consists of everything in H times a.
    Then this argument above is true?
     
  2. jcsd
  3. Feb 16, 2010 #2

    Fredrik

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    That's not an argument, it's just the definition of the cosets. :smile:
     
  4. Feb 16, 2010 #3
    My doubt is whether there is a counter-example such that when aH and bH are distinct left cosets, Ha and Hb are not distinct right cosets, because this statement looks suspicious.
     
  5. Feb 16, 2010 #4

    Fredrik

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    When H is a normal subgroup, it's very easy to prove that the statement is true, but I don't see why it must be true when H isn't a normal subgroup. So if you're looking for counterexamples, start by thinking of examples of subgroups that aren't normal.
     
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