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 from Grigorii Riabov about a 1991 theorem by Moorhouse, 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. More precisely, the result appears as Proposition 4.1 in
G.E. Moorhouse, Bruck nets, codes, and characters of loops, Designs, Codes and Cryptography 1 (1991), No. 1, 7–29
However, I can't quite follow the fine details of the proof: among other things. First, I find it a bit sketchy in places. Second, the proposition refers to $3$-nets instead ofrather than simple graphs, and to quasigroupsloops (called loopsedit: in the papersee Peter Taylor's comment below) instead ofrather than groups — more precisely, Moorhouse assumes that $G_1$ is a group, but allows $G_2$ to be a loop.
This leads me to the following questions:
- Q1. Is there any "modern reference" for Moorhouse's theorem (with all the necessarya more recent source — perhaps a book — with additional details foror a more complete treatment, even if only in the proofspecial case where both $G_1$ and $G_2$ are groups (edit: see YCor's comment below)?
- Q2. Does the theorem also hold for infinite groups? If not, what breaks down?
