Automorphisms of the Two-Dimensional Linear Groups Over Integral Domains

1974 ◽  
Vol 96 (1) ◽  
pp. 1 ◽  
Author(s):  
Martin H. Dull
Keyword(s):  
Linear Groups ◽  
Two Dimensional ◽  
Integral Domains

Two-Dimensional Linear Groups Over Local Rings

1969 ◽  
Vol 21 ◽  
pp. 106-135 ◽  
Author(s):  
Norbert H. J. Lacroix
Keyword(s):  
Local Ring ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Linear Groups ◽  
Local Rings ◽  
Two Dimensional ◽  
Normal Subgroups ◽  
Field Case

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .

Free quotients of infinite rank of GL2 over Dedekind domains

1999 ◽  
Vol 129 (1) ◽  
pp. 77-84
Author(s):  
A. W. Mason
Keyword(s):  
Factor Group ◽  
Krull Dimension ◽  
Two Dimensional ◽  
Integral Domains ◽  
Dimension Zero ◽  
Infinite Rank ◽  
Dedekind Domains ◽  
Free Quotient

This paper is concerned with integral domains R, for which the factor group SL2(R)/U2(R) has a non-trivial, free quotient, where U2(R) is the subgroup of GL2(R) generated by the unipotent matrices. Recently, Krstić and McCool have proved that SL2(P[x])/U2(P[x]) has a free quotient of infinite rank, where P is a domain which is not a field. This extends earlier results of Grunewald, Mennicke and Vaserstein.Any ring of the type P[x] has Krull dimension at least 2. The purpose of this paper is to show that result of Krstić and McCool extends to some domains of Krull dimension 1, in particular to certain Dedekind domains. This result, which represents a two-dimensional anomaly is the best possible in the following sense. It is well known that SL2(R) = U2(R), when R is a domain of Krull dimension zero, i.e. when R is a field. It is already known that for some arithmetic Dedekind domains A, the factor group SL2(A)/U2(A) has a free quotient of finite (and not infinite) rank.

Homomorphisms of two-dimensional linear groups over Laurent polynomial rings and Gaussian domains

1991 ◽  
Vol 109 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Yu Chen
Keyword(s):  
Commutative Ring ◽  
Linear Group ◽  
Unit Group ◽  
Commutative Rings ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Linear Groups ◽  
Two Dimensional ◽  
Euclidean Domain ◽  
Invertible Matrices

Let GL2(R) be the general linear group of 2 × 2 invertible matrices in M2(R) over a commutative ring R with 1 and SL2(R) be the special linear group consisting of 2 × 2 matrices over R with determinant 1. In this paper we determine the homomorphisms from GL2 and SL2, as well as their projective groups, over Laurent polynomial rings to those groups over Gaussian domains, i.e. unique factorization domains (cf. Theorems 1, 2, 3 below). We also consider more generally the homomorphisms of non-projective groups over commutative rings containing a field which are generated by their units (cf. Theorems 4 and 5). So far the homomorphisms of two-dimensional linear groups over commutative rings have only been studied in some specific cases. Landin and Reiner[7] obtained the automorphisms of GL2(R), where R is a Euclidean domain generated by its units. When R is a type of generalized Euclidean domain with a degree function and with units of R and 0 forming a field, Cohn[3] described the automorphisms of GL2(R). Later, Cohn[4] applied his methods to the case of certain rings of quadratic integers. Dull[6] has considered the automorphisms of GL2(R) and SL2(R), along with their projective groups, provided that R is a GE-ring and 2 is a unit in R. McDonald [9] examined the automorphisms of GL2(R) when R has a large unit group. The most recent work of which we are aware is that of Li and Ren[8] where the automorphisms of E2(R) and GE2(R) were determined for any commutative ring R in which 2, 3 and 5 are units.

scholarly journals On the finite two dimensional linear groups

1968 ◽  
Vol 10 (4) ◽  
pp. 419-435 ◽  
Author(s):  
Peter Lorimer
Keyword(s):  
Linear Groups ◽  
Two Dimensional
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