A structure theorem for KT-nearfields
All known examples of sharply 3-transitive permutation groups G contain an abelian subgroup fixing one point and acting regularly on the remaining points. Hence G can be written as \(G=\{(x\to a+mx)\), \((x\to a+m \sigma (b+x)) |\) a,b,m\(\in F\), \(m\neq 0\}\) for a suitable KT-near- field (F,\(\sigma)\), i.e. a near-field F (with distributive law \(x(y+z)=xy+xz)\) with an automorphism \(\sigma\) of the multiplicative group of F satisfying the functional equation \(\sigma (1+\sigma (x))=1-\sigma (1+x).\) The author proves the following result: for every KT-near-field (F,\(\sigma)\) with char(F)\(\neq 2\), the kernel \(K=\{k\in F |\) \((x+y)k=xk+yk\) for every x,y\(\in F\}\) of F is a commutative field, and \(\sigma (k)=k^{-1}\) for every \(k\in K^*.\) This generalizes results of \textit{W. Benz} and \textit{S. Elliger} [ibid. 1, 267-274 (1968; Zbl 0165.172)] and \textit{E. Hartmann} [Mitt. Math. Ges. Hamb. 10, 757-762 (1980; Zbl 0562.12021)]. If char(F)\(\neq 0\) and char(F)\(\not\equiv 1\) (mod 4), then the prime field of F is contained in the center of F; examples shown that the restrictions on the characteristic of F cannot be relaxed.
scientific article; zbMATH DE number 3975054