# Thread: If and only if proof involving cosets.

1. ## 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.

2. Originally Posted by mjlaz
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.

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