Let there be an Abelian group with a binary operation $\ast$ on a set $S$. Let such a group respect the following propriety:
$$ (X\ast Y)\ast Y = X$$
For any $X$ and $Y$ in $S$. I realize that by using the associative propriety we can say:
$$ (X\ast Y)\ast Y = X \;\therefore\; X\ast (Y\ast Y) = X \;\therefore\; Y\ast Y = e$$
Where $e$ is the identity element of $\ast$, meaning that under $\ast$, any element of $S$ is it's own inverse. Could such a group exist? If so, what kind of proprieties would it have? Are there any examples?