A 2-local characterization of \(M_{12}\) (Q1409593)

This article is one in a series of papers providing simplified, modern, character free and computer free treatments of the existence and uniqueness of the sporadic simple groups and of the normalizers of subgroups of prime order and of Sylow subgroups in these groups. Here, a \(2\)-local characterization -- particularly suitable for the classification of the quasithin finite simple groups of even characteristic -- of the Mathieu group \(M_{12}\) is established: Theorem 1: Let \(G\) be a finite group, \(z\) an involution in \(G\), \(H=C_G(z)\), \(Q=O_2(H)\), and \(X\in\text{Syl}_3(H)\). Assume (a) \(Q\) is extraspecial of order \(32\), (b) \(H/Q\cong S_3\) and \(C_Q(X)=\langle z\rangle \), and (c) \(z\) is not weakly closed in \(Q\) with respect to \(G\). Then one of the following holds: (1) There is a normal \(E_8\)-subgroup \(V\) of \(G\) with \(G/V\cong L_3(2)\). (2) \(G\cong A_8\) or \(A_9\), and the two \(Q_8\)-subgroups of \(Q\) are not normal in \(H\). (3) \(G\cong M_{12}\), and the two \(Q_8\)-subgroups of \(Q\) are normal in \(H\). Apart from the proof of Theorem 1, it is shown that \(M_{12}\) satisfies the assumptions of Theorem 1 and lists of the normalizers of subgroups of prime order and of Sylow subgroups of \(M_{11}\) and \(M_{12}\) together with proofs for their correctness are given.