Groups that are almost homogeneous
type: Item
instance of: scholarly article
title: Groups that are almost homogeneous
zbMATH Open document ID: 0890.20025
DOI: 10.1023/A:1005090519480
author: Helmut Mäurer
author: Markus Stroppel
published in: Geometriae Dedicata
publication date: 1998-05-12T00:00:00Z
review text: A group \(G\) is called (almost) homogeneous if the group \(\Aut G\) has at most (3 orbits) 2 orbits on \(G\). It is easy to see that the homogeneous abelian groups are exactly the additive groups of vector spaces. A commutative field \(F\) is called \(q\)-suited if the cyclotomic polynomial \(\Phi_q=(x^q-1)/(x-1)\) has \(q-1\) different roots in \(F\), and \(\Aut F\) acts transitively on the set of these roots. If \(F\) is a field, then \(\Delta(q,F,d)\) denotes the semidirect product of the additive group \(U\) of a \(d\)-dimensional vector space over \(F\) and \(\Omega=\{f\in F\mid f^q=1\}\) acting on \(V\) by scalar multiplication. Let \(F\) be a commutative field, let \(V\) and \(Z\) be vector spaces over \(F\), and consider a symplectic bilinear map \(\beta\colon V\times V\to Z\) (that is \((v,v)=0\) for each \(v\in V\)). Let \(GH(V,Z,\beta)\) be the set \(V\times Z\) endowed with the operation \((v,y)\circ(w,z)=(v+w,y+(v,w)+z)\). Then \(GH(V,Z,\beta)\) is a group and the operation \(\circ\) coincides with addition on each one-dimensional subspace of \(V\times Z\). The group \(GH(V,Z,\beta)\) is called a generalized Heisenberg group. Let \(G\) be an almost homogeneous group. Then (a) If the group \(G\) is abelian then it is the additive group either of a vector space or of a free module over the local ring \(Z(p^2)\) for some prime \(p\). (b) If the group \(G\) is a solvable torsion group of composite exponent then it is isomorphic to one of the groups \(\Delta(q,F,d)\) for some \(q\)-suited field \(F\). (c) If the group \(G\) is nilpotent of exponent \(p\) then it is a generalized Heisenberg group. (d) If the group \(G\) is nilpotent of exponent \(p^2\) then either it is abelian, or \(p=2\) and \(G\) is special with \(G/Z(G)\) isomorphic to \(Z(G)\).
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 1098484
zbMATH Keywords: automorphisms of groups
zbMATH Keywords: Heisenberg groups
zbMATH Keywords: Suzuki \(2\)-groups
zbMATH Keywords: almost homogeneous groups
zbMATH Keywords: additive groups of vector spaces
zbMATH Keywords: solvable torsion groups
zbMATH Keywords: nilpotent groups
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
full work available at URL: a:1005090519480
OpenAlex ID: W1525743940
description: scientific article; zbMATH DE number 1098484