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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 \(H=H_ d\triangleleft H_{d- 1}\triangleleft\cdots\triangleleft H_ 0=G\) of subgroups of \(G\). In this case write \(H\triangleleft^ dG\).
Let \(A\) be an associative ring with 1. For any subgroup \(H\) of \(GL_ nA\), let \(L(H)\) be the largest ideal \(B\) of \(A\) such that \(E_ n(A,B)\subset H\), and let \(J(H)\) denote the least ideal \(B\) such that \(H\subset G_ n(A,B)\), where \(G_ n(A,B)\) is the inverse image of the center of \(GL_ n(A/B)\) under the canonical homomorphism \(GL_ nA\to GL_ n(A/B)\). 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 \(sr(S^{-1}A)\leq n-1\); (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 \(x_ i\) of A such that \(x_ 1+\cdots+x_ N=1\) and \(1+ax_ i\in GL_ 1A\) for all \(i\).
The author proves that, if \(H\) is a subgroup of \(GL_ nA\) such that \(H\triangleleft^ dGL_ nA\) (\(n\geq 3\) and \(d\geq 1\)), then \(J(H)^ m\subset L(H)\) for \(m=(4^ d-1)/3\); conversely, if \(H\) is a subgroup of \(E_ nA\) such that \(J(H)^ m\subset L(H)\) (\(n\geq 3\) and \(m\geq 1\)), then \(H\triangleleft^{m+1}E_ nA\triangleleft GL_ nA\). In particular, a subgroup \(H\) of \(E_ nA\) (\(n\geq 3)\) is subnormal if and only if \(J(H)^ m\subset L(H)\) for some \(m\geq 1\).
Reviewer: Li Fuan

MSC:

20H25 Other matrix groups over rings
20G35 Linear algebraic groups over adèles and other rings and schemes
20E15 Chains and lattices of subgroups, subnormal subgroups
19B10 Stable range conditions
20E07 Subgroup theorems; subgroup growth
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