An improved 3-local characterization of McL and its automorphism group. (Q402422)

The authors give a \(3\)-local characterization of the McLaughlin sporadic simple group McL and its automorphism group \(\Aut(\mathrm{McL})\). As a matter of fact, this result extends earlier work of \textit{C. Parker} and \textit{P. Rowley} [J. Algebra 319, No. 4, 1752-1775 (2008; Zbl 1140.20015)] in which McL and \(\Aut(\mathrm{McL})\) are characterized by certain \(3\)-local information.NEWLINENEWLINE Main Theorem of this article (Theorem 1.1): Suppose that \(G\) is a finite group, \(S\in\mathrm{Syl}_3(G)\), \(Z=Z(S)\) and \(J\) is an elementary abelian subgroup of \(S\) of order \(3^4\). Further assume that (i) \(O^{3'}(N_G(Z))\approx 3_+^{1+4}.2\cdot\mathrm{Alt}(5)\); (ii) \(O^{3'}(N_G(J))\approx 3^4.\mathrm{Alt}(6)\); and (iii) \(C_G(O_3(C_G(Z)))\leqslant O_3(C_G(Z))\). Then \(G\cong\mathrm{McL}\) or \(\Aut(\mathrm{McL})\).NEWLINENEWLINE The difference between this theorem and the characterization theorem of Parker and Rowley is as follows. In fact, the main theorem of the former article essentially assumes that the group under investigation is of local characteristic 3, but the main theorem here only assumes that the group \(G\) has parabolic characteristic \(3\).