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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 $Sp\sb FR$ (resp. $Gp\sb FR)$ 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 $\tau (e,u,x)\in Sp\sb FR$ 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 $Ep\sb FR$ be the subgroup of $Gp\sb FR$ generated by all $\tau$ (e,u,x) with unimodular vectors e of V. In this paper, the author describes all subgroups of $Gp\sb FR$ which are normalized by $Ep\sb FR$. 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 $r\sp 2b$, $2a+a\sp 2r\in B$ 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 $Ep\sb F(A,B)$ of $Sp\sb FR$ and $Gp\sb F(A,B)$ of $Gp\sb FR$. Under some conditions on V and F, the author proves that a subgroup H of $Gp\sb FR$ is normalized by $Ep\sb FR$ if and only if $Ep\sb F(A,B)\subset H\subset Gp\sb F(A,B)$ for a symplectic ideal (A,B) of R.
Reviewer: E.Abe

MSC:
20G35Linear algebraic groups over adèles and other rings and schemes
20H25Other matrix groups over rings
20E07Subgroup theorems; subgroup growth
20H05Unimodular groups, congruence subgroups (matrix groups)
WorldCat.org
Full Text: DOI
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