Local finiteness of periodic sharply triply transitive groups.
The finite sharply 3-transitive permutations groups were classified by H. Zassenhaus in the 1930's. If \(G\) is a sharply 3-transitive finite group acting on a set \(\Omega\), then \(|\Omega|=p^n+1\), with \(p\) a prime number and there is an automorphism of \(F=\text{GF}(p^n)\) with \(\omega^2=1\) (\(\omega\) can be the identity) such that \(G\) is isomorphic to the group of the maps of the projective line \(F\cup\{\infty\}\) defined by \[ x\mapsto\frac{a\cdot\omega^*(x)+b}{c\cdot\omega^*(x)+d}\qquad a,b,c,d\in F,\quad ad\neq bc \] where \(\omega^*(x)=x\) if \(ad-bc\) is a square in \(F^\times\) and \(\omega^*(x)=\omega(x)\) otherwise. The Zassenhaus result has been generalized by \textit{O. H. Kegel} to locally finite groups [Arch. Math. 18, 337-348 (1967; Zbl 0189.31103)]. The main result in the present paper is Theorem 2: If \(G\) is a sharply 3-transitive group with periodic stabilizer of two points, then \(G\) is locally finite. This result is obtained by proving that the stabilizer of a point admits a non-trivial elementary abelian normal \(p\)-subgroup for some prime number \(p\). An important consequence of Theorem 2 is that every periodic sharply 3-transitive group is locally finite. It should be emphasized that the construction of the first example of a sharply 3-transitive group whose point stabilizers have no non-trivial abelian normal subgroup has been announced by Katrin Tent in 2015.
scientific article; zbMATH DE number 6485743
scientific article