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Thread: If and only if proof involving cosets.

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    If and only if proof involving cosets.

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

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

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

    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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