Cyclic quotients of transitive groups
type: Item
instance of: scholarly article
title: Cyclic quotients of transitive groups
zbMATH Open document ID: 0974.20001
published in: Journal of Algebra
publication date: 2001-09-04T00:00:00Z
review text: In this note two interesting theorems about finite permutation groups and an application to the Galois theory of function fields resp. algebraic curves are proved. Theorem 1. If \(A\) is a transitive subgroup of the symmetric group \(S_n\) on \(n\) letters and \(G\) is a normal subgroup of \(A\) such that \(A/G\) is cyclic, then \(|A/G|\leq n\) and \(|A/G|<n\) if \(G\) is transitive. Theorem 3. If \(A\) is a finite primitive permutation group then a point stabilizer in \(A\) has a faithful orbit. The formulation of the mentioned application is more complicated (see Theorem 2 and sections 5 and 6 of the paper). The proofs are based on using a new group invariant \(\mu(H)\) related to central chief factors in normal series of the group \(H\). The classification of finite simple groups is used for obtaining the full results. However, some partial results are proved without invoking the classification.
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 20B20
zbMATH DE Number: 1565872
zbMATH Keywords: finite permutation groups
zbMATH Keywords: transitive permutation groups
zbMATH Keywords: normal subgroups
zbMATH Keywords: cyclic factor groups
zbMATH Keywords: cyclic quotients
zbMATH Keywords: primitive groups
zbMATH Keywords: point stabilizers
zbMATH Keywords: suborbits
zbMATH Keywords: Galois theory
zbMATH Keywords: coverings of curves
zbMATH Keywords: procyclic fields
zbMATH Keywords: extensions of constants
author: Robert Guralnick
MaRDI profile type: Publication
full work available at URL: jabr.2000.8540
OpenAlex ID: W2008386260
cites work: On the maximal subgroups of the finite classical groups
cites work: On Abelian Quotients of Primitive Groups
cites work: Maximal subgroups of finite groups
cites work: Q3684278
cites work: The Local Maximal Subgroups of Exceptional Groups of Lie Type, Finite and Algebraic
cites work: FIXED POINT RATIOS IN EXCEPTIONAL GROUPS OF RANK AT MOST TWO
cites work: On the minimal degree of a primitive permutation group
cites work: Q4779764
cites work: Abhyankar's conjecture on Galois groups over curves
cites work: The maximal subgroups of the finite 8-dimensional orthogonal groups \(P\Omega ^ +_ 8(q)\) and of their automorphism groups
cites work: Q3996618
cites work: An order bound for the point stabilizer of a primitive permutation group
cites work: Finite permutation groups with large Abelian quotients
cites work: Fixed point ratios in actions of finite exceptional groups of Lie type.
cites work: Q3791327
cites work: Maximal subgroups of exceptional groups of Lie type, finite and algebraic
cites work: Subgroups of Maximal Rank in Finite Exceptional Groups of Lie Type
cites work: Coverings of the affine line in characteristic \(p>0\) and Abhyankar's conjecture
cites work: Galois groups of unramified covers of projective curves in characteristic \(p\)
cites work: Q5691146
DOI: 10.1006/JABR.2000.8540
description: scientific article; zbMATH DE number 1565872