Non-abelian representations of some sporadic geometries
For a point-line geometry \(G = (P,L)\) with exactly three points per line, (the non-abelian) universal representation group is defined by \(R(G) = \langle z_p, p \in P \mid z^2_p = 1\) for \(p \in P\), \(z_p z_q z_r = 1\) for \(\{p, q, r\} \in L\rangle\). The commutator factor group \(V\) of \(R(G)\) is a vector space over \(GF(2)\), giving a nice module for \(G\), provided that \(V\) is not trivial. The authors prove the following main result: if \(G\) is the 2-local parabolic geometry of the sporadic simple group \(F_1\) (the monster) or of \(F_2\) (the baby monster), then \(R(G) \cong F_2\) or \(2 \cdot F_2\), respectively. The authors also determine the universal representation groups of some geometries related to the Mathieu groups \(M_{22}\), \(M_{24}\) and to the Conway groups \(Co_1\) and \(Co_2\).
scientific article; zbMATH DE number 883772