Each invertible sharply \(d\)-transitive finite permutation set with \(d\geq 4\) is a group
type: Item
instance of: scholarly article
title: Each invertible sharply \(d\)-transitive finite permutation set with \(d\geq 4\) is a group
zbMATH Open document ID: 0971.20001
DOI: 10.1023/A:1011211907282
author: Pasquale Quattrocchi
author: A. Bonisoli
published in: Journal of Algebraic Combinatorics
publication date: 2001-10-29T00:00:00Z
review text: 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.
reviewed by: Alice Devillers
Mathematics Subject Classification ID: 20B20
Mathematics Subject Classification ID: 51D20
Mathematics Subject Classification ID: 20B40
zbMATH DE Number: 1563244
zbMATH Keywords: invertible sharply \(d\)-transitive permutation sets
zbMATH Keywords: permutation groups
zbMATH Keywords: MAGMA
zbMATH Keywords: point-block geometries
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1563244