Verbal embeddings and a geometric approach to some group presentations
type: Item
instance of: scholarly article
title: Verbal embeddings and a geometric approach to some group presentations
zbMATH Open document ID: 0679.20029
DOI: 10.1016/0021-8693(89)90132-4
author: Thomas A. Fournelle
author: Kenneth W. Weston
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: For a ring R, the Steinberg group St(n,R) is the group generated by symbols \(x_{i,j}(r)\), \(1\leq i,j\leq n\), subject to the relations that \(x_{i,j}(r)x_{i,j}(s)=x_{i,j}(r+s)\), \([x_{i,j}(r),x_{j,k}(s)]=x_{i,k}(rs)\) for \(i\neq k\), and some commuting relations. The authors consider a generalization called a linkage group. A linkage graph \(\Gamma\) consists of a collection of vertices and a collection of ordered triples of vertices, called linkages. The linkage group L(R,\(\Gamma)\) is generated by symbols \(r_ v\), for all \(r\in R\) and \(v\in V(\Gamma)\) subject to the relations that \(r_ vs_ v=(r+s)_ v\) and that \([r_ u,s_ w]=(rs)_ v\) whenever (u,v,w) is a linkage. A (special) verbal embedding of R into a group G is a function \(\phi\) : \(R\to G\) such that \(\phi (r)\phi (s)=\phi (r+s)\) and the commutator \([a^{-1}\phi (r)a,b^{-1}\phi (s)b]=\phi (rs)\), where a and b are two fixed elements of G that generate a subgroup C of G, called the constant group. The authors prove that for every H with \(C\leq H\leq G\), there exists a linkage graph \(\Gamma_ H\) and an epimorphism from \(L(R,\Gamma_ H)\) to the subgroup of G generated by the H-conjugates of the subgroup generated by \(a^{-1}\phi (R)a\), \(\phi\) (R), and \(b^{- 1}\phi (R)b\). In particular, if G is simple, then taking \(H=G\) shows that G is a homomorphic image of a linkage group. The authors study linkage groups with the property that for each \(v\in V(\Gamma)\), the function sending r to \(r_ v\) is injective. Assuming that R has a (multiplicative) identity element and no zero divisors, this implies that for any linkage (u,v,w), the subgroup of G is isomorphic to the group U(R) of \(3\times 3\) upper triangular matrices with entries in R and all diagonal entries equal to 1. Moreover, it forces very strong restrictions on \(\Gamma\) ; in particular, it cannot contain certain kinds of subgraphs. This leads to various consequences; for example, C must have order at least 6, and if C is abelian…
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 4112875
zbMATH Keywords: Cayley
zbMATH Keywords: linkage graph
zbMATH Keywords: linkage group
zbMATH Keywords: generated by symbols
zbMATH Keywords: verbal embedding
zbMATH Keywords: upper triangular matrices
zbMATH Keywords: Steinberg groups
zbMATH Keywords: \(A_ n\)
zbMATH Keywords: Mathieu groups
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90132-4
OpenAlex ID: W2048045358
cites work: Q3684278
cites work: On algebraic embeddings of rings into groups
cites work: On finite simple groups of characteristic 2 type
cites work: On nilpotent class 2 groups and the Steinberg groups ST(3,R)
Recommended article: Q3349085
Recommended article: On normal verbal embeddings of groups
Recommended article: Q3770716
Recommended article: Q3645582
Recommended article: Q3784300
Recommended article: On normal verbal embeddings of some classes of groups.
Recommended article: On geometric embedding problems and semiabelian groups
Recommended article: On some residual properties of the verbal embeddings of groups
Recommended article: Geometric group presentations
Recommended article: Q4294554
description: scientific article; zbMATH DE number 4112875