Isomorphisms of special sharply 3-transitive permutation groups (Q1193243)

In this paper the isomorphisms of sharply 3-transitive groups are characterized for the class of sharply 3-transitive groups described by structures called special KT-fields. All examples of sharply 3-transitive groups known to the author are constructed by such special KT-fields. It is shown that if \((F,\varepsilon)\) and \((\widehat F,\widehat\varepsilon)\) are special KT-fields and \(T_ 3(F,\varepsilon)\) and \(T_ 3(\widehat F,\widehat\varepsilon)\) the corresponding sharply 3-transitive groups, then every isomorphism from \(T_ 3(F,\varepsilon)\) onto \(T_ 3(\widehat F,\widehat\varepsilon)\) is of the form \(\sigma\to(\tau\delta)\sigma(\tau\delta)^{-1}\), where \(\tau\in T_ 3(\widehat F,\widehat\varepsilon)\) and \(\delta\) is an isomorphism of \((F,\varepsilon)\) onto \((\widehat F,\widehat\varepsilon)\). In particular the automorphism group of \(T_ 3(F,\varepsilon)\) is isomorphic to the semidirect product \(T_ 3(F,\varepsilon)\cdot\text{Aut}(F,\varepsilon)\). These results are also extended to the class of sharply 3-transitive groups whose elements of order three have one fixed point (type 1) or three fixed points (type 3) and whose associated KT-field is a KT-near- field.