In the present paper we show that comparing results of Hyman Bass [15] with those of Oleg Tavgen [10], one immediately gets the following result, which is both more general, and more precise than all recent results pertaining to unitriangular factorisations.
Unitriangular factorization is a presentation of a linear group as a product of unipotent radicals of a Borel subgroup and its opposite. Whether this decomposition is known for Chevalley groups over rings of stable rank 1 and some Dedekind rings of arithmetic type, the case of twisted groups has been studied only over finite fields. In the present paper we give a much simpler proof for twisted groups over finite fields and the field of complex numbers.
We formulate and prove relative versions of several decompositions known in the theory of Chevalley groups over commutative rings. These decompositions are used to obtain factorizations in terms of subsystem subgroups of type A ℓ and upper estimates of the width of principal congruence subgroups with respect to Tits-Vaserstein generators.Theorem 2. Assume that sr(I) 2. Then the group Epin(2ℓ, R, I) = E(D ℓ , R, I) is a product of at most 9 regularly embedded subgroups of type A ℓ−1 .As another application, we obtain results on the bounded generation of the relative elementary group. Recall from [16, Theorem 2] that E(Φ, R, I) can be generated by the set of Stein-Tits-Vaserstein generators Date: October 2, 2018.where the J-shape of a root β ∈ Φ is defined by the formula: shape(J, β) = i∈J m i (β)α i . Clearly, ∆ J is a reductive subset, while S ± J and Σ ± J are parabolic and special subsets, respectively. For two disjoint subsets I, J ⊆ Π one has Σ ± I∪J = Σ ± I ∪ Σ ± J , ∆ I∪J = ∆ I ∩ ∆ J . We omit curly braces in the above notations when J is a one-or two-element set, e. g. ∆ k = ∆ {k} and Σ ± i,j = Σ ± {i,j} , etc.Lemma 2.1 ([1, Lemma 1]). Let α, β ∈ Σ ± J be a pair of roots of the same length such that shape(J, α) = shape(J, β) = 0. Then α and β are conjugate under the action of W (∆ J ).
An explicit and elementary proof is given to the fact that Suzuki and Ree groups can be decomposed into the product of 4 of their Sylow psubgroups, where p is the defining characterictic.
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