Unitriangular Factorization of Twisted Chevalley Groups

Abstract

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.

Minimal length factorizations of finite simple groups of Lie type by unipotent Sylow subgroups

“…When specializing their results to elementary Chevalley groups over finite fields, they get that any non-twisted finite simple group of Lie type is a product of four unipotent Sylows. Later on, these results were extended by Smolensky in [11] to cover some twisted Chevalley groups over finite fields or the field of complex numbers.…”

“…Thus, the results in [14], [15], [16], [13], [11] and [12], combine to give a different proof of the four Sylow claim of Theorem 1.…”

Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

“…After the completion of our proof of Theorem 1.2, and in parallel to its publication in preprint form ( [18]), Smolensky made available a preprint in which he shows that every Suzuki and Ree group is a product of four unipotent Sylow subgroups ( [30]). Thus, the results in [32], [33], [34], [29] and [30], combine to give a different proof of the four Sylow claim of Theorem 1.2. Theorem 1.2 and the cp-factorization of finite simple groups of Lie type by Borel subgroups mentioned above clearly motivate a general study of cp-factorizations of a finite group G by subgroups which are nilpotent or solvable.…”

Products of Sylow Subgroups in Suzuki and Ree Groups

“…A slightly more technical case of 2 A 2n is elaborated in [10] by explicitly factorizing the group of type 2 A 2 .…”

“…The proof is based on Tavgen rank reduction theorem [Tav90,Tav92], which doesn't care about the base ring at all, and the decomposition of SL 2 , which can be carried over any ring of stable rank 1. A slightly more technical case of 2 A 2n is elaborated in [Smo13] by explicitly factorizing the group of type 2 A 2 .…”