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Normal subgroups of symplectic groups over rings. (English) Zbl 0669.20040

Let R be a commutative ring with 1 and F be an alternating form on an R- module V. The symplectic group \(Sp_ FR\) (resp. \(Gp_ FR)\) is the group of all automorphisms of the R-module V which preserve (resp. multiply) the form F. For every \(e,u\in V\) such that \(F(e,u)=0\) and any \(x\in R\), define \(\tau (e,u,x)\in Sp_ FR\) by \(\tau (e,u,x)v=v+uF(e,u)+eF(u,v)+exF(e,v)\) for all \(v\in V\). A vector v of V is called F-unimodular if \(F(V,v)=R\). Let \(Ep_ FR\) be the subgroup of \(Gp_ FR\) generated by all \(\tau\) (e,u,x) with unimodular vectors e of V. In this paper, the author describes all subgroups of \(Gp_ FR\) which are normalized by \(Ep_ FR\). This generalizes many previous results of Dickson, Klingenberg, Abe, Bak et al. The author defines a symplectic ideal of R as a pair (A,B), where A is an ideal of R and B is an additive subgroup of A such that \(r^ 2b\), \(2a+a^ 2r\in B\) for all \(r\in R\), \(b\in B\) and \(a\in A\). Generalizing the congruence subgroups, corresponding to a symplectic ideal (A,B), there correspond normal subgroups \(Ep_ F(A,B)\) of \(Sp_ FR\) and \(Gp_ F(A,B)\) of \(Gp_ FR\). Under some conditions on V and F, the author proves that a subgroup H of \(Gp_ FR\) is normalized by \(Ep_ FR\) if and only if \(Ep_ F(A,B)\subset H\subset Gp_ F(A,B)\) for a symplectic ideal (A,B) of R.
Reviewer: E.Abe

MSC:

20G35 Linear algebraic groups over adèles and other rings and schemes
20H25 Other matrix groups over rings
20E07 Subgroup theorems; subgroup growth
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
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