A block-theory-free characterization of \(M_{24}\) (Q917691)

In Section 5 of [\textit{D. Held}, J. Algebra 13, 253-296 (1969; Zbl 0182.043)], the following characterization of the simple group \(M_{24}\) was given. Suppose that G is a finite simple group containing a central involution \(z_ 1\) such that \(H_ 1=C_ G(z_ 1)\) is isomorphic to the centralizer of a central involution in \(M_{24}\). Here \(H_ 1\) contains two elementary abelian normal subgroups of order 16: E and \(E_ 1\). In order to distinguish \(M_{24}\) from the simple groups \(L_ 5(2)\) and He, assume further that \(N_ G(E_ 1)=H_ 1\) and that \(N_ G(E)/E\cong A_ 8\). In the reference above, the local structure of G was developed to a point where the Thompson group order formula could be applied to conclude that \(| G| =| M_{24}|\) and then the Ph.D. thesis of R. Stanton was applied to force \(G\cong M_{24}\). In this paper, the authors further develop the local structure of G to obtain for G a J. A. Todd system of generators and relations for \(M_{24}\). Thus the authors conclude that \(G\cong M_{24}\) without utilizing the work of R. Stanton.