On tactical decompositions of class number 2 (Q1922805)

Let \({\mathcal P}_1, {\mathcal P}_2\) and \({\mathcal B}_1, {\mathcal B}_2\) be decompositions of the point set \({\mathcal P}\) and the block set \({\mathcal B}\) of a symmetric design. This decomposition is called tactical (of class number \(2\)) if the number of points in \({\mathcal P}_i\) on a block of \({\mathcal B}_j\) does not depend on the choice of the block, but only on \(i\) and \(j\). Dually, the number of blocks in \({\mathcal B}_i\) through a point of \({\mathcal P}_j\) is independent of the choice of the point. If \(| {\mathcal P}_1| =| {\mathcal B}_1| =1\), the decomposition is called of affine type. A decomposition is symmetric if \(| {\mathcal P}_i| =| {\mathcal B}_i| \), \(i=1,2\). In the paper under review, the authors investigate (symmetric) tactical decompositions. For instance, they show that symmetric designs of prime order admit only tactical decompositions of affine type. Moreover, the paper contains a classification of symmetric tactical decompositions of finite projective planes (under the additional assumption that the tactical configuration \(({\mathcal P}_1, {\mathcal B}_1)\) is a divisible design).