On the normal subgroups of sharply 2- and sharply 3-transitive permutation groups
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 \(Q=\langle x\sigma(x^{-1})\mid x\in F^*\rangle\)).
scientific article; zbMATH DE number 1146259
scientific article