A construction of the Held simple group
This article contains a new result within the larger frame of constructing sporadic simple groups by embedding an amalgam into some linear group, equivalently, a geometry into some vector space. Here, the sporadic simple group \(He\) discovered by Held is constructed as subgroup of \(\text{GL}(51,11)\). For this purpose, first matrices in \(\text{GL}(51,11)\) for the two maximal parabolics \(2^{1+6}L_3(2)\) and \(2^6.3S_6\) of the tilde geometry are constructed. Then, a subgroup \(X\cong S_4(4)\) is identified in the subgroup \(G\leq\text{GL}(51,11)\) generated by these matrices and a graph on the set of conjugates of \(X\) in \(G\) is defined. Considering the action of \(G\) on this graph, it is proved that \(|G|=|He|\). Finally, \(G\) is identified with \(He\) by proving that it satisfies the original definition of \(He\) by \textit{D. Held} [cf. J. Algebra 13, 253-296 (1969; Zbl 0182.04302)].
scientific article; zbMATH DE number 1565884