Local finiteness of periodic sharply triply transitive groups.
type: Item
instance of: scholarly article
title: Local finiteness of periodic sharply triply transitive groups.
zbMATH Open document ID: 1330.20003
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Algebra and Logic
publication date: 2015-09-28T00:00:00Z
review text: 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.
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20B22
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 6485743
zbMATH Keywords: sharply transitive permutations groups
zbMATH Keywords: periodic permutations groups
zbMATH Keywords: periodic groups
zbMATH Keywords: locally finite groups
zbMATH Keywords: periodic two-point stabilizers
MaRDI profile type: Publication
full work available at URL: s10469-015-9321-1
OpenAlex ID: W2047206786
cites work: Q3822294
cites work: Q3253828
cites work: Zur Struktur lokal endlicher Zassenhausgruppen
cites work: The Kourovka notebook. Unsolved problems in group theory.
cites work: Q4528894
cites work: Q4349305
cites work: Q3148501
cites work: Fixed-point-free 2-finite automorphism groups.
cites work: On the Shunkov groups acting freely on Abelian groups.
cites work: Bemerkungen über Fastbereiche und scharf zweifach transitive Gruppen. (Remarks on near-domains and sharply 2-transitive groups.)
cites work: On periodic groups with an almost regular involution
cites work: Groups with an almost-regular involution
cites work: Q4745979
DOI: 10.1007/S10469-015-9321-1
Recommended article: Sharply 3-transitive Groups with Finite Element
Recommended article: Sharply 3-transitive groups.
Recommended article: \textit{KT}-fields and sharply triply transitive groups
Recommended article: Q3363335
Recommended article: On sharply k-transitive permutation sets (k\(\geq 3)\)
Recommended article: Sharply 2-transitive groups with point stabilizer of exponent 3 or 6
Recommended article: Q3148501
Recommended article: A structure theorem for KT-nearfields
Recommended article: A note on groups generated by involutions and sharply 2-transitive groups
Recommended article: Embeddings of infinite permutation groups in sharp, highly transitive, and homogeneous groups
description: scientific article; zbMATH DE number 6485743