zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The subnormal structure of general linear groups over rings. (English) Zbl 0711.20030
The subnormal structure of general linear groups over rings has been studied by many people. The author of the present paper has obtained the best result. A subgroup $H$ of a group $G$ is called subnormal if there is a finite chain $H=H\sb d\triangleleft H\sb{d- 1}\triangleleft\cdots\triangleleft H\sb 0=G$ of subgroups of $G$. In this case write $H\triangleleft\sp dG$. Let $A$ be an associative ring with 1. For any subgroup $H$ of $GL\sb nA$, let $L(H)$ be the largest ideal $B$ of $A$ such that $E\sb n(A,B)\subset H$, and let $J(H)$ denote the least ideal $B$ such that $H\subset G\sb n(A,B)$, where $G\sb n(A,B)$ is the inverse image of the center of $GL\sb n(A/B)$ under the canonical homomorphism $GL\sb nA\to GL\sb n(A/B)$. Assume $A$ satisfies one of the following conditions: (1) for every maximal ideal $P$ of the center $C$ of $A$ there is a multiplicative subset $S$ in $C-P$ such that $sr(S\sp{-1}A)\le n-1$; (2) $A/rad(A)$ is von Neumann regular; (3) for every element $a$ of $A$ there is a natural number $N$ and there are elements $x\sb i$ of A such that $x\sb 1+\cdots+x\sb N=1$ and $1+ax\sb i\in GL\sb 1A$ for all $i$. The author proves that, if $H$ is a subgroup of $GL\sb nA$ such that $H\triangleleft\sp dGL\sb nA$ ($n\ge 3$ and $d\ge 1$), then $J(H)\sp m\subset L(H)$ for $m=(4\sp d-1)/3$; conversely, if $H$ is a subgroup of $E\sb nA$ such that $J(H)\sp m\subset L(H)$ ($n\ge 3$ and $m\ge 1$), then $H\triangleleft\sp{m+1}E\sb nA\triangleleft GL\sb nA$. In particular, a subgroup $H$ of $E\sb nA$ ($n\ge 3)$ is subnormal if and only if $J(H)\sp m\subset L(H)$ for some $m\ge 1$.
Reviewer: Li Fuan

20H25Other matrix groups over rings
20G35Linear algebraic groups over adèles and other rings and schemes
20E15Chains and lattices of subgroups, subnormal subgroups
19B10Stable range conditions ($K_1$)
20E07Subgroup theorems; subgroup growth
Full Text: DOI