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Math Help - If and only if proof involving cosets.

  1. #1
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    If and only if proof involving cosets.

    Suppose that G is a group and that H is a subgroup of G. Show that for a,b \in G that aH = bH if and only if b^{-1}a \in H.

    Is there a trick to this one? My scratch work is leading nowhere.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mjlaz View Post
    Suppose that G is a group and that H is a subgroup of G. Show that for a,b \in G that aH = bH if and only if b^{-1}a \in H.

    Is there a trick to this one? My scratch work is leading nowhere.
    You would agree that since e\in H that a\in aH right? But, since as sets aH=bH we must have that a\in bH but every element of bH is of the form bh,\text{ }h\in H so that a=bh\implies b^{-1}a=h\in H

    You do the other way.
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    Of course it makes sense now. Thanks a lot.
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    I feel like I've made the same argument twice in the reverse direction. What do you think? Here is my work (I typed it up in LaTeX): http://i.imgur.com/e3LaW.jpg

    -- Marc
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mjlaz View Post
    I feel like I've made the same argument twice in the reverse direction. What do you think? Here is my work (I typed it up in LaTeX): http://i.imgur.com/e3LaW.jpg

    -- Marc
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