M, B and Co1 are recognisable by their prime graphs

Abstract

Abstract The prime graph, or Gruenberg–Kegel graph, of a finite group 𝐺 is the graph Γ ⁢ ( G ) \Gamma(G) whose vertices are the prime divisors of | G | \lvert G\rvert and whose edges are the pairs { p , q } \{p,q\} for which 𝐺 contains an element of order p ⁢ q pq . A finite group 𝐺 is recognisable by its prime graph if every finite group 𝐻 with Γ ⁢ ( H ) = Γ ⁢ ( G ) \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, M \mathrm{M} , the Baby Monster, B \mathrm{B} , and the first Conway group, Co 1 \mathrm{Co}_{1} . We prove that these three groups are recognisable by their prime graphs.