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Let  p  be a prime integer and letℤ  p{\mathbb{Z}_{p}}    be the ring of  p -adic integers. By a purely computational approach we prove that each nonzero normal element of a noncommutative Iwasawa algebra over the special linear groupSL  3   ⁢   (   ℤ  p   ){\mathrm{SL}_{3}(\mathbb{Z}_{p})}    is a unit. This gives a positive answer to an open question in [F. Wei and D. Bian, Erratum: Normal elements of completed group algebras overSL  n   ⁢   (   ℤ  p   )\mathrm{SL}_{n}(\mathbb{Z}_{p})    [mr2747414], Internat. J. Algebra Comput. 23 2013, 1, 215] and makes up for an earlier mistake in [F. Wei and D. Bian, Normal elements of completed group algebras overSL  n   ⁢   (   ℤ  p   )\mathrm{SL}_{n}(\mathbb{Z}_{p})   , Internat. J. Algebra Comput. 20 2010, 8, 1021–1039] simultaneously.

Normal elements of noncommutative Iwasawa algebras over SL 3 (ℤ_ p )