n-Cube as a Cayley Graph
The question is asking you to find a suitable group $\Gamma$ and a subset $S$, so that $CG(\Gamma,S)$ is the n-cube.

Since by definition of $CG$, the nodes are the elements of $\Gamma$, you don't have a choice here, leaving the group operation and $S$.

$xy$ and the inverse of $y$ are then understood with respect to the group operation you choose.

Also be aware that the identity element being in $\Gamma$ does not imply that all other vertices are connected with it, since the edges are determined by $S$, not $\Gamma$.