$\text{M}$, $\text{B}$ and $\text{Co}_1$ are recognisable by their prime graphs

Abstract

The prime graph, or Gruenberg--Kegel graph, of a finite group $G$ is the graph $\Gamma(G)$ whose vertices are the prime divisors of $|G|$, and whose edges are the pairs $\{p,q\}$ for which $G$ contains an element of order $pq$. A finite group $G$ is recognisable by its prime graph if every finite group $H$ with $\Gamma(H)=\Gamma(G)$ is isomorphic to $G$. By a result of Cameron and Maslova, every such group must be almost simple, so one natural case to investigate is that in which $G$ is one of the $26$ sporadic simple groups. Existing work of various authors answers the question of recognisability by prime graph for all but three of these groups, namely the Monster, $\text{M}$, the Baby Monster, $\text{B}$, and the first Conway group, $\text{Co}_1$. We prove that these three groups are recognisable by their prime graphs.