Baer subgeometry partitions
Let \(n\geq 3\) be an integer and let \(q\) be a prime power. It is known that the projective \((n-1)\)-space \(PG(n-1,q^2)\) can be partitioned into pairwise disjoined subgeometries isomorphic to \(PG(n-1,q)\) precisely when \(n\) is odd [see \textit{J. W. P. Hirschfeld}, ``Projective geometries over finite fields,'' 2nd ed., Oxford Univ. Press (1998; Zbl 0899.51002)]. Such a partition is called a Baer subgeometry partition, abbreviated by BSP. Now let \(n\) be odd, \(n\geq 3\). Then there exists a BSP of \(PG(n-1,q^2)\) that can be obtained by taking point orbits under an appropriate subgroup of a Singer cycle of \(PG(n-1,q^2)\) (see Hirschfeld). If this is true then the BSP is called classical. The aim of the paper is to contruct BSP's that are nonclassical. One of the results is the following. If \(n\geq 3\), \(n\) odd and if \(q^n>8\) then nonclassical BSP's of \(PG(n-1,q^2)\) do exist (Corollary 3.2). The authors point out that there are close connections between BSP's and several areas of combinatorial interest, such as flag-transitive translation planes.
scientific article; zbMATH DE number 1443419