Automorphisms of permutational wreath products
type: Item
instance of: scholarly article
title: Automorphisms of permutational wreath products
zbMATH Open document ID: 0651.20030
DOI: 10.1016/0021-8693(88)90121-4
author: Fletcher Gross
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: The main result of this paper is the following theorem: Let P be a transitive permutation group on the finite set I, \(| I| >1\), let H be the stabilizer in P of a point in I, and S the number of orbits of H on I. Let R be a finite nonabelian indecomposable group. Let \(G=R Wr (P,I)\) and let A be the group of all automorphisms of G which fix the subgroup \(R^ I\). Then Inn(G)\(\leq A\leq Aut(G)\) and \[ | A/Inn(G)| =| Hom(H,Z(R)| | Out(R)| | N_{Aut(P)}(H)/Inn_ H(P)| | Hom(R,Z(R)|^{s-1}. \] Furthermore, \(A=Aut(G)\) if any of the following is satisfied: (1) \(H=1\) and either \(| I| >2\) or R is not a special dihedral group. (2) \(| I| >2\) and H is contained in some core-free maximal subgroup of P. (3) \(O_ 2(P)=1\). (4) R is not of dihedral type. By using the result certain characteristic subgroups and \(| Out(G)|\) are given in the present paper.
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20B05
zbMATH DE Number: 4061502
zbMATH Keywords: transitive permutation group
zbMATH Keywords: stabilizer
zbMATH Keywords: number of orbits
zbMATH Keywords: automorphisms
zbMATH Keywords: characteristic subgroups
reviewed by: Piroska Lakatos
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90121-4
OpenAlex ID: W2081471947
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description: scientific article; zbMATH DE number 4061502