Let $G$ be a group and $H$ be a subgroup of $G$. I need to show that for a subset $X$ of $G$, if we let $X^{-1}$ denote the set of elements inverse to elements of $X$ - i.e., that $X^{-1}=\{ x^{-1} | x \in X \}$ - then $X$ is a left coset of $H$ in $G$ iff $X^{-1}$ is a right coset of $H$ in $G$.
My attempt so far is this:
$(\implies)$ Suppose $X$ is a left coset of $H$ in $G$. Then, $X = xH = \{ xy | y \in H\}$.
Since $H$ is a subgroup of $G$, if $xy \in X$, $xyy^{-1} \in X$, since $xyy^{-1} = x(id) = x \in X$.
I know it's not a lot to go on, but that's pretty much all I have so far.
I don't really know where to go from there as far as making a right coset of the form $Hx^{-1}$ pop out.
And I have thought a little bit about the $(\Longleftarrow)$ direction and figure it must be very similar, but I'm sort of stuck on the same thing with it.
Could somebody please let me know where I should go from here?
Thanks.
Just consider that a left coset is of the form $$ x H = \{ x h : h \in H \}. $$ now what is $(x h)^{-1}$? And what is $\{ h^{-1} : h \in H \}$?