Some properties of inductively minimal geometries (Q1281131)

An inductively minimal geometry is a geometry \(\Gamma\) of rank \(n\) with a connected diagram, together with a flag transitive automorphism group \(G\) such that for every residue \(R\) of rank \(r\) with a connected subdiagram, the order of the induced automorphism group \(G_R\) is at most equal to \((r+1)!\). The authors prove some properties of such geometries. For instance, \(\Gamma\) satisfies the intersection property and every rank 2 residue is either a partial linear space (two different elements are incident with at most one common element) or a generalized digon (all elements of one type are incident with all elements of the other type). Other properties in terms of (quasi)maximal subgroups and primitive actions of stabilizers of flags are given. These inductively minimal geometries appear to be most helpful and important for the geometries of the symmetric groups of maximal rank.