A characterization of the Fischer group \(M(23)\) by a 2-local subgroup
The author proves the following result: Theorem: Let \(G\) be a finite simple group with \(O(C_G(i))=1\) for every involution \(i\) in \(G\). Suppose \(G_{11}\) possess an elementary abelian subgroup \(E\) of order 2 such that the subgroup \(M=N_G(E)\) has the following properties. (i) \(C_G(E)=E\). (ii) \(N_G(E)/E\cong M_{23}\). Then \(G=N_G(E)\) or \(G\cong M(23)\). In proving this the author first shows that \(M_{23}\) can act on an 11-dimensional \(F_2\)-vector space in essentially two different ways. For one module one has that a central involution \(z\in E\) has the property \(C_G(Z)\subseteq N_G(E)\). Goldschmidt's fusion result eventually gives a contradiction. In the case of the other module one can find a class \(e^G\) of involutions with \(e^G\cap N_G(E)\subseteq E\). In controlling how products of involutions of \(E\) are distributed in \(G\)-classes one can establish that \(e^G\) is a class of odd transpositions.
scientific article; zbMATH DE number 3558067