Normal elements in the mod-𝑝 Iwasawa algebra over SL𝑛(ℤ𝑝): A computational approach

Abstract

Abstract This is the last in a series of articles where we are concerned with normal elements of noncommutative Iwasawa algebras over SL n ⁢ ( ℤ p ) {\mathrm{SL}_{n}(\mathbb{Z}_{p})} . Our goal in this portion is to give a positive answer to an open question in [D. Han and F. Wei, Normal elements of noncommutative Iwasawa algebras over SL 3 ⁢ ( ℤ p ) \mathrm{SL}_{3}(\mathbb{Z}_{p}) , Forum Math. 31 2019, 1, 111–147] and make up for an earlier mistake in [F. Wei and D. Bian, Normal elements of completed group algebras over SL n ⁢ ( ℤ p ) \mathrm{SL}_{n}(\mathbb{Z}_{p}) , Internat. J. Algebra Comput. 20 2010, 8, 1021–1039] simultaneously. Let n ( n ≥ 2 {n\geq 2} ) be a positive integer. Let p ( p > 2 {p>2} ) be a prime integer, ℤ p {\mathbb{Z}_{p}} the ring of p-adic integers and 𝔽 p {\mathbb{F}_{p}} the finite filed of p elements. Let G = Γ 1 ⁢ ( SL n ⁢ ( ℤ p ) ) {G=\Gamma_{1}(\mathrm{SL}_{n}(\mathbb{Z}_{p}))} be the first congruence subgroup of the special linear group SL n ⁢ ( ℤ p ) {\mathrm{SL}_{n}(\mathbb{Z}_{p})} and Ω G {\Omega_{G}} the mod-p Iwasawa algebra of G defined over 𝔽 p {\mathbb{F}_{p}} . By a purely computational approach, for each nonzero element W ∈ Ω G {W\in\Omega_{G}} , we prove that W is a normal element if and only if W contains constant terms. In this case, W is a unit. Also, the main result has been already proved under “nice prime” condition by Ardakov, Wei and Zhang [Non-existence of reflexive ideals in Iwasawa algebras of Chevalley type, J. Algebra 320 2008, 1, 259–275; Reflexive ideals in Iwasawa algebras, Adv. Math. 218 2008, 3, 865–901]. This paper currently provides a new proof without the “nice prime” condition. As a consequence of the above-mentioned main result, we observe that the center of Ω G {\Omega_{G}} is trivial.