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Math Help - Set Containment Problem

  1. #1
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    Abstract Algebra Problem

    I see there is an old string addressing this question, but I need to prove it using set containment both directions. Not sure where to start.

    a, b are in G, a group. H is a subgroup. If aH = bH then Ha^(-1) = Hb^(-1).
    Last edited by GoldendoodleMom; April 16th 2008 at 06:44 AM.
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  2. #2
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    Quote Originally Posted by GoldendoodleMom View Post
    I see there is an old string addressing this question, but I need to prove it using set containment both directions. Not sure where to start.

    a, b are in G, a group. H is a subgroup. If aH = bH then Ha^(-1) = Hb^(-1).
    You can do this easily if you write the conditions clearly and think reverse.

    To prove: Ha^{-1} \subset Hb^{-1}
    That is to prove: \forall h \in H, ha^{-1}b \in H
    But that is the same as proving a^{-1}b \in H
    Now since, aH = bH \Rightarrow a^{-1}b \in H, we are done.
    The other way round is exactly similar except for different variable name.
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