Characterization of chain geometries of finite dimension by their automorphism group
The author considers so called chain spaces, i.e. incidence structures with the following properties: (i) any three pairwise incident points are in a unique block, (ii) the residue of any point is a partial affine space, (iii) any block contains at least three points and any point is in some block. A chain space is called strong if (i) for any two points x and y there is a third point z such that z is incident with both x and y, (ii) for any point p the blocks in the residue are fairly large (Let C be a block \(D_ p\) the set of points incident with p and \(C_ p=C\cap D_ p\), then \(| C\setminus C_ p| +1<| C_ p|).\) Let K be a commutative field and R be a K-algebra. Then there is a natural chain space \(\Sigma\) (K,R) on the projective line where one starts with the block \(\{\) R(s,t), s,t\(\in K\), (s,t)\(\neq (0,0)\}\), and the blocks are the images of this block under the projective group \(\Gamma\) (R). The author proves a number of results which basically classify this chain space among the chain spaces possessing large automorphism groups. To give a flavour of the result here is one on strong spaces: Let \(\Sigma\) be a strong chain space such that the residual spaces have as underlying spaces an affine space of dimension at least three. Suppose that \(\Sigma\) possesses an automorphism group \(\Gamma\) acting sharply transitive on the triples of pairwise incident points. Suppose finally that whenever \(a\in \Gamma\) interchanges two incident points than a is forced to be an involution. Then \(\Sigma\) is \(\Sigma\) (K,R) for some pair K, R (R commutative) and \(\Gamma\cong \Gamma (R)\).
scientific article; zbMATH DE number 4180211