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

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). 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 SL₃(ℤ_p), Forum Math. 31 (2019), no. 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), Internat. J. Algebra Comput. 20 (2010), no. 8, 1021–1039] simultaneously. Let n (n ≥ 2) be a positive integer. Let p (p > 2) be a prime integer, ℤ_p the ring of p-adic integers and F_p the finite filed of p elements. Let G = Γ₁(SL_n(ℤ_p)) be the first congruence subgroup of the special linear group SL_n(ℤ_p) and Ω_G the mod-p Iwasawa algebra of G defined over F_p. By a purely computational approach, for each nonzero element W ∈ Ω_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), no. 1, 259–275; Reflexive ideals in Iwasawa algebras, Adv. Math. 218 (2008), no. 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 is trivial.