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Sandwich classification for \(O_{2n+1}(R)\) and \(U_{2n+1}(R,\Delta)\) revisited. (English) Zbl 1405.20043

Let \(R\) be a commutative ring and let \(O_{2n+1}(R)\) be the odd-dimensional orthogonal group, where \(n \geq1\). The elementary elements \(T(x)\), \(x \in R\), are by definition unipotent matrices of simple type. They generate the elementary subgroup \(EO_{2n+1}(R)\). The most important result in the first part of this paper is a technical theorem. The author proves that, for every \(\sigma=(\sigma_{ij}) \in O_{2n+1}(R)\), each \(T(x)\), where \(x=\sigma_{ij}\;\mathrm{or}\; \sigma_{ii}-\sigma_{jj}\), \(i \neq j\), is a product of \(N\), say, \(EO_{2n+1}(R)\)-conjugates of \(\sigma^{\pm 1}\), where \(N \leq 48\) or \(N=64n+148\). A pair \((I,J)\) of \(R\)-ideals which are closely related in a specified way is called admissible. Such a pair is used to define two subgroups (i) \(EO_{2n+1}(R,I,J)\), the elementary subgroup of level \((I,J)\) and (ii) \(C_{2n+1}(R,I,J)\), the full congruence subgroup of level \((I,J)\). Using the above, he proves the following sandwich classification theorem (SCT):
A subgroup \(H\) of \(O_{2n+1}(R)\) is normalized by \(EO_{2n+1}(R)\) if and only if \[ EO_{2n+1}(R,I,J) \leq H \leq CO_{2n+1}(R,I,J), \] for some admissible pair \((I,J)\).
The author then uses a similar approach to prove, more generally, an SCT for the much more complicated case of the odd-dimensional unitary group \(U_{2n+1}(R, \Delta)\), where \((R,\Delta)\) is a Hermitian form ring. Although these results are already known the author’s approach leads to much shorter and simpler proofs.
The simplest version of an SCT applies to \(\mathrm{GL}_n(R)\), where \(n \geq 3\). Here, the \(R\)-ideals alone are sufficient to classify the \(E_n(R)\)-normalized subgroups. The first case of this result (indeed the first example of any SCT) is is due to H. Bass [Publ. Math., Inst. Hautes Étud. Sci. 22, 489–544 (1964; Zbl 0248.18025)]. It should be noted that, in general, sandwich classification theorems of the above type only hold for linear groups if the dimension is sufficienly high. It is known, for example, that the \(\mathbb{Z}\)-ideals fail completely to classifiy in this way the \(E_2(\mathbb{Z})\)-normalized subgroups of \(\mathrm{GL}_2(\mathbb{Z})\).

MSC:

20G35 Linear algebraic groups over adèles and other rings and schemes
20H25 Other matrix groups over rings

Citations:

Zbl 0248.18025
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References:

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