Sandwich classification for O _2n+1(R) and U _2n+1(R,Δ) revisited

Abstract

In a recent paper, the author proved that if n ě 3 is a natural number, R a commutative ring and σ P GLnpRq, then t kl pσij q where i ‰ j and k ‰ l can be expressed as a product of 8 matrices of the form ǫ σ˘1 where ǫ P EnpRq. In this article we prove similar results for the odd-dimensional orthogonal groups O2n`1pRq and the odd-dimensional unitary groups U2n`1pR, ∆q under the assumption that R is commutative and n ě 3. This yields new, short proofs of the Sandwich Classification Theorems for the groups O2n`1pRq and U2n`1pR, ∆q.

On products of conjugacy classes in general linear groups

“…Assume that i j+1 > j + 2. Then τ j+1,i j+1 = 0 by (5). Hence det σ 1,3,i 3 ...,i k 1,3,i 3 ...,i k = det(A) det(E).…”

“…We will show by induction on j that Q(j) holds for any 3 ≤ j ≤ k. j = 3 Assume that i 3 > 4. Then τ 3i 3 = 0 by (5). But τ 31 = 0 by P (3), whence det(σ 1,3,i 3 ...,i k 1,3,i 3 ...,i k ) − det(τ 1,3,i 3 ...,i k 1,3,i 3 ...,i k ) = τ 23 det(σ 3,i 3 ...,i k 1,i 3 ...,i k ) = 0.…”

“…In 1960, J. Brenner [3] showed that in the case R = Z there is a bound for the number of E n (R)-conjugates needed for such an expression. In 2018, the author [5] proved that indeed for any commutative ring there is a bound for the number of E n (R)-conjugates needed for such an expression. In 2020, the author [6] obtained bounds for different classes of noncommutative rings.…”

Reverse decomposition of unipotents over noncommutative rings I: General linear groups

“…provided n is large enough with respect to the stable rank of R. Here E n (R) denotes the elementary subgroup, E n (R, I) the relative elementary subgroup of level I and C n (R, I) the full congruence subgroup of level I (cf. [8]). Bass's result, which is one of the central points in the structure theory of general linear groups, is known as Sandwich Classification Theorem.…”

Groups normalized by the odd unitary group

“…Later these results were generalized for unitary Bak's groups in [2,26] and even for odd Petrov's groups in [6]. Also Raimund Preusser gave new short proofs of the sandwich classification of normal subgroups in the case of commutative rings in his recent papers [25,27].…”