Some chain geometries determined by transformation groups
The action of a permutation group \(G\) on a set \(X\) naturally extends componentwise to an action of \(G^n\) on \(X^n\). The authors define the geometry of chains over the permutation group \((G,X)\) as the geometry \(\prod^n(G,X)\) whose point set is \(X^n\) and whose blocks are the images under \(G^n\) of the diagonal base block \(\Delta_n=\{(x,\ldots,x)\in X^n:x\in X\}\). This geometry generalises the notion of a Minkowski plane and weak Minkoswki space in case \(G\) is sharply 3-transitive on \(X\); for \(n=2\) one obtains a \(B^*\) geometry, that is, a Minkowski plane without the axiom of touching, and the case \(n\geq 2\) was investigated by \textit{H.-J. Samaga} [J. Geom. 12, 69--87 (1979; Zbl 0391.51006)]. Furthermore, \(\prod^n(G,X)\) has an interpretation as a curve space as defined by \textit{A. Herzer} [Results Math. 12, 278--288 (1987; Zbl 0638.51007)]. The authors investigate combinatorial properties of the geometries of chains and their automorphism groups. The geometry \(\prod^n(G,X)\) has \(n\) parallelisms, two points in \(X^n\) being \(i\)-parallel for \(1\leq i\leq n\) if and only if their \(i\)th coordinates are equal. Furthermore, two points are called parallel if they are \(i\)-parallel for some \(i\). It is shown that \(n\) mututally distinct and parallel points, that is, a clique in the the graph \(\Gamma\) with vertex set \(X^n\) and adjacency given by parallelity, are \(i\)-parallel for some \(i\) if and only if any points that are parallel to all the \(n\) points are parallel themselves. In particular, the \(i\)-parallel classes can be recovered from \(\Gamma\). The authors verify that the group generated by the symmetric group \(S_n\) (acting as a permutation group on the factors of \(X^n\)), the group \(G^n\) and the automorphism group of the permutation group \((G,X)\) acting diagonally on \(X^n\) is a group of automorphisms of \(\prod^n(G,X)\) and various subgroups thereof are investigated. Furthermore, if \(| X| \geq n\) and \(G\) is 2-transitive, then this group is the full autonmorphism group of the chain geometry.
scientific article; zbMATH DE number 2163346
scientific article