Each invertible sharply \(d\)-transitive finite permutation set with \(d\geq 4\) is a group
A permutation set \(H\) on \(X\) is a subset of \(\text{Sym}(X)\). \(H\) is called invertible if \(H\) contains the identity and the inverse \(h^{-1}\) of every permutation \(h\) in \(H\). Let \(H\) be an invertible sharply \(d\)-transitive permutation set on a finite set \(X\) with \(d\geq 4\). The authors prove that \(H\) must be a permutation group. More precisely: (i) if \(d\geq 6\), \(H\) is either \(S_d\), \(S_{d+1}\) or \(A_{d+2}\), (ii) if \(d=5\), \(H\) is either \(S_5\), \(S_6\), \(A_7\) or \(M_{12}\), (iii) if \(d=4\), \(H\) is either \(S_4\), \(S_5\), \(A_6\) or \(M_{11}\). The proof relies at some points on a complete investigation by computer (with MAGMA programs). The authors make a link between permutation sets and some point-block geometries. If the permutation set is sharply \((m+2)\)-transitive, the corresponding geometry is called a Minkowski \(m\)-structure.
invertible sharply \(d\)-transitive permutation sets
scientific article; zbMATH DE number 1563244
scientific article