Given a group $G$, let $\Gamma(G)$ be the simple graph with vertex set $G \times G$, in which two distinct vertices $(x, y)$ and $(u, v)$ are adjacent if and only if either $x = u$, or $y = v$, or $xy = uv$.

I have recently learned about a theorem by Moorhouse (from the 1990s), stating that two finite groups $G_1$ and $G_2$ are isomorphic if and only if the graphs $\Gamma(G_1)$ and $\Gamma(G_2)$ are isomorphic. This leads me to the following questions:

• Q1. What is a reference (with all necessary details) for Moorhouse's theorem?
• Q2. Does the theorem also hold for infinite groups? If not, what breaks down?