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The  prime graph , or  Gruenberg–Kegel graph , of a finite group  is the graphΓ⁢(G)\Gamma(G)whose vertices are the prime divisors of|G|\lvert G\rvertand whose edges are the pairs{p,q}\{p,q\}for which  contains an element of orderp⁢qpq. A finite group  is  recognisable  by its prime graph if every finite group  withΓ⁢(H)=Γ⁢(G)\Gamma(H)=\Gamma(G)is isomorphic to . By a result of Cameron and Maslova, every such group must be almost simple, so one natural case to investigate is that in which  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,Co1\mathrm{Co}_{1}. We prove that these three groups are recognisable by their prime graphs.

M, B and Co 1 are recognisable by their prime graphs