Recognisability of the sporadic groups by the isomorphism types of their prime graphs

Abstract

The prime graph of a finite group $G$ is the labelled graph $Γ(G)$ with vertices the prime divisors of $|G|$ and edges the pairs $\{p,q\}$ for which $G$ contains an element of order $pq$. A group $G$ is recognisable by its prime graph if every group $H$ with $Γ(H)=Γ(G)$ is isomorphic to $G$. Cameron and Maslova have shown that every group that is recognisable by its prime graph is almost simple, which justifies the significant amount of attention that has been given to determining which simple (or almost simple) groups are recognisable by their prime graphs. This problem has been completely solved for certain families of simple groups, including the sporadic groups. A natural extension of the problem is to determine which groups are recognisable by their unlabelled prime graphs, i.e. by the isomorphism types of their prime graphs. There seem to be only very limited results in this direction in the literature. Here we determine which of the sporadic finite simple groups are recognisable by the isomorphism types of their prime graphs. We also show that for every sporadic group $G$ that is not recognisable by the isomorphism type of $Γ(G)$, there are infinitely many groups $H$ with $Γ(H) \cong Γ(G)$.