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

Abstract

Abstract. We prove that every finite simple group G of Lie type satisfies G = U U − U U − where U is a unipotent Sylow subgroup of G and U − is its opposite. We also characterize the cases for which G = U U − U . These results are best possible in terms of the number of conjugates of U in the above factorizations.

Products of Sylow Subgroups in Suzuki and Ree Groups

“…Immediately after the publication of the preprint of the present article, the preprint of another article, [6] appeared, containing a proof of the same result as our main Theorem. The proof goes along the same line, with the following difference:…”

“…It is now clear that the shortest and simplest proof would be as follows: 1) Tavgen rank reduction theorem; 2) Lemma 2.7 of [6]; and 3) the part of Lemmas 1 and 2 dealing with the lift of the nontrivial element of the Weyl group, and the same proof for A 1 and 2 A 2 . Apart from the decomposition into a product of Sylow subgroups (alias unitriangular factorization), the articles [4,6] also study a decomposition into a product of the conjugates of the Borel subgroup, which is also known as the Gauss decomposition [11].…”

“…Apart from the decomposition into a product of Sylow subgroups (alias unitriangular factorization), the articles [4,6] also study a decomposition into a product of the conjugates of the Borel subgroup, which is also known as the Gauss decomposition [11]. We now list some remarks concerning their generalizations:…”

Products of conjugacy classes in simple algebraic groups in terms of diagrams

“…Unipotent factorization of G. The following result is due to Vavilov, Smolensky and Sury [28]. A proof can also be obtained with the method used in [6] for finite groups of Lie type. We refer to [6] and the references therein for more background on this result.…”

“…A proof can also be obtained with the method used in [6] for finite groups of Lie type. We refer to [6] and the references therein for more background on this result.…”