André planes and nests of reguli
In this interesting paper, the authors determine all translation planes that admit certain collineation groups. Sometimes only the order of the group is considered. For instance, let \(\pi\) be a translation plane of order \(q^ n\) with kernel containing GF(q) that admits a linear collineation of order \([(q^ n-1)/(q-1)]^ 2.\) The authors show that if \[ ((q^ n-1)/(q-1),\prod^{n-1}_{i=1}(q^ i-1))=1, \] then \(\pi\) is an André plane. More typical of the results found in the paper are the following two: (1) Let \(\pi\) be a translation plane of order \(q^ n\), \(n>2\), with kernel GF(q). If \(\pi\) admits an abelian collineation group G of order \(((q^ n-1)/(q-1))^ 2\) in the linear translation complement, then \(\pi\) is a generalized André plane. (2) If \(\pi\) is a translation plane of order \(q^ n\) with kernel GF(q) that admits an abelian collineation group G of order \((q^ n-1)^ 2\) in the linear translation complement, then \(\pi\) is an André nearfield plane. The reviewer [Eur. J. Comb. 10, No.4, 319-330 (1989; Zbl 0691.05009)] constructed a family of translation planes of order \(q^ 2\) and kernel GF(q) by replacing a \((q+1)\)-nest of reguli. In general, a t-nest of reguli is a collection of t reguli in a regular spread so that every line of the spread is contained in zero or two reguli of the collection. In the paper under review the following result is proven: Let \(\pi\) be a translation plane of order \(q^ 2\) with kernel containing GF(q) that admits a collineation group G in the translation complement which is the direct product of two cyclic homology groups of order \(q+1\) with affine axes. Then either (i) \(\pi\) is André, (ii) q is odd and constructed from a Desarguesian plane \(\Sigma\) by a nest replacement of a \((q+1)\)- nest of reguli in \(\Sigma\), or (iii) q is odd and constructed from a Desarguesian plane \(\Sigma\) by a nest replacement of a \((q+1)\)-nest and the replacement of a set of mutually disjoint André nests in \(\Sigma\). Quite surprisingly, if \(\pi\) is a translation plane of order \(q^ n\), \(n>2\), satisfying the above hypotheses (except that the cyclic homology groups have order \((q^ n-1)/(q-1),\) then the authors show \(\pi\) is necessarily an André plane.
scientific article; zbMATH DE number 4194579