I'm taking a class on graph theory that uses "Graph Theory (Graduate Texts in Mathematics)" by Bondy and Murty. One of the questions is about Cayley graphs and the n-cube, and I don't understand how to interpret it. It runs as follows:
Let $\Gamma$ be a group and $S$ be a subset of $\Gamma$ not including the identity element. Suppose that the inverse of every element in $S$ also belongs to $S$. The Cayley graph of $\Gamma$ with respect to $S$ is the graph $CG(\Gamma, S)$ with vertex set $\Gamma$ in which two vertices $x$ and $y$ are adjacent iff $xy^{-1}\in S$.
Okay. I follow so far.
Recall that the n-cube is the graph whose vertex set is the set of all n-tuples of 0s and 1s, where two n-tuples are adjacent if they differ in precisely one coordinate.
Makes sense.
Show that the n-cube is a Cayley graph.
What does it mean to talk about "$xy$" when $x$ and $y$ are n-tuples? What is the inverse of an n-tuple?
Someone I asked about the problem suggested that I treat $\Gamma$ here as the additive group $({\mathbb Z}/2{\mathbb Z})^n$, and so take $xy^{-1}$ to mean elementwise subtraction of $y$ from $x$, mod 2. But then it seems like $(0, 0, ..., 0)\in\Gamma$ and, since it's the identity element, every vertex will have an edge connecting to it, and that isn't what the n-cube looks like. Googling, I also see that there is an interpretation of tuples as nested sets, but then I don't see how the product of two nested sets would ever be in S, since it will have a different cardinality from either of the original tuples. Interpreting the tuples as vectors can't work either since then $xy^{-1}$ will have different dimensions than either of the original tuples.
What is this question asking?