A characterization of \(M_{22}\)
This paper, using modern techniques, presents a simplified local characterization of the Mathieu simple group \(M_{22}\). The main result is the following: Let \(G\) be a finite group with an involution \(z\) such that \(H=C_G(z)\) contains a normal extraspecial group \(Q\) of order 32 such that \(Z(Q)=\langle z\rangle\), \(H/Q\cong Z_2\times S_3\), \(C_H(Q)\leq Q\), \(z^G\cap Q\neq\{z\}\) and \(C_Q(X)=\langle z\rangle\) for any \(X\in\text{Syl}_3(H)\). Choosing \(g\in G-H\) such that \(z^g\in Q\), it is known that \(Q\cap Q^g\) is elementary abelian of order 4 or 8. If also \(Q\cap Q^g\) is of order 4, then say that \(G\) is of type \(M_{22}\). Main Theorem. If \(G\) is of type \(M_{22}\) then either \(G\cong M_{22}\) or \(G\) is isomorphic to the split extension of \(E_{16}\) by \(A_6\).
scientific article; zbMATH DE number 1014044
scientific article