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

TL;DR

This paper proves that the Monster, Baby Monster, and Conway group Co1 are uniquely identifiable by their prime graphs, completing the recognition problem for all sporadic simple groups.

Contribution

The paper establishes that the three remaining sporadic simple groups are recognisable solely by their prime graphs, filling a gap in the classification.

Findings

Monster group is recognisable by its prime graph

Baby Monster group is recognisable by its prime graph

Conway group Co1 is recognisable by its prime graph

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.