On the normal subgroups of sharply 2- and sharply 3-transitive permutation groups
type: Item
instance of: scholarly article
title: On the normal subgroups of sharply 2- and sharply 3-transitive permutation groups
zbMATH Open document ID: 0961.20004
DOI: 10.1007/BF01237505
published in: Journal of Geometry
publication date: 1998-06-23T00:00:00Z
review text: The purpose of this paper is to give a survey about the normal subgroups of sharply 2-transitive permutation groups \(T_2\) and sharply 3-transitive groups \(T_3\). Essential use is made of the fact that the \(T_2\) can be uniquely described as the group of linear transformations of a neardomain \((F,+,\cdot)\) and \(T_3\) as the group of fractional linear transformations of a uniquely determined \(KT\)-field \((F,\varepsilon)\). It is shown that \(T_2\) and \(T_3\) have a least normal subgroup if \(F\), resp. \((F,\varepsilon)\) are nearfields or \(\text{char }F\neq 2\). This smallest normal subgroup of \(T_3\) is simple if \((F,\varepsilon)\) is a nearfield or contains a suitable central element. All known examples satisfy this condition, since up to now no proper neardomain which is not a nearfield is known. Also it is shown that there exists a bijection between the nontrivial normal subgroups of \(T_2\) (resp. \(T_3\)) with all normal subgroups of the multiplicative group \(F^*:=F\setminus\{0\}\) containing the set \(D:=\langle d_{a,b}\mid a,b\in F\rangle\) (resp. the set \(D\cdot Q\) with \(Q=\langle x\sigma(x^{-1})\mid x\in F^*\rangle\)).
reviewed by: Heinrich Wefelscheid
Mathematics Subject Classification ID: 20B22
Mathematics Subject Classification ID: 12K05
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1146259
zbMATH Keywords: normal subgroups
zbMATH Keywords: sharply 2-transitive permutation groups
zbMATH Keywords: sharply 3-transitive permutation groups
zbMATH Keywords: groups of linear transformations
zbMATH Keywords: neardomains
zbMATH Keywords: \(KT\)-fields
zbMATH Keywords: nearfields
author: Heinz Waehling
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1146259