On the maximal rank of primitive residually connected geometries for \(M_{22}\)
The authors show that there are no flag transitive residual connected and primitive geometries \(\Gamma\) for \(G= M_{22}\) of rank of least 7. In fact, there is one of rank 6. Let \(n\) be the rank of \(\Gamma\) and \(F\) be a flag of rank \(n\). Identify the flag with the set \(\{1,\dots,n\}\). Then the assumptions about \(\Gamma\) yield that for all \(I,J\subseteq\{1,\dots,n\}\) with \(I\subseteq J\) and \(|J\setminus I|= 1\) we have that \(G_J\) is maximal in \(G_I\). In particular, \(G_i\) is a maximal subgroup of \(G\) for all \(i\in\{1,\dots,n\}\). Hence the proof is a careful study of the maximal subgroups of \(G\) and maximal subgroups of various intersections.
scientific article; zbMATH DE number 6138247