Thread: If and only if proof involving cosets.

1. 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.

2. Originally Posted by mjlaz
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.

3. Of course it makes sense now. Thanks a lot.

4. 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

5. Originally Posted by mjlaz
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