Partial Augmentations in Integral Group Rings of GL_n(k): Hertweck's Question Fails as Printed for n ≥ char k; the Restricted Questions Remain Open

In the problem list of the 2007 Oberwolfach mini-workshop "Arithmetik von Gruppenringen", M. Hertweck asked: for G = GL_n(k), k a finite field of characteristic p, and π: V(ℤG) → G induced by the inclusion G ⊆ M_n(k), does every torsion unit u of V(ℤG) have a nonzero partial augmentation at the conjugacy class of π(u)? We show that the question as printed has a negative answer for every n ≥ p, by an elementary example, while the restricted questions that were probably intended, for units of order prime to p and for n < p, remain open in general; for them we prove partial results. The counterexamples are units u = x + (x − 1)hx̂ of order p from V. Bovdi's family in Problem 4 of the same list, rationally conjugate to the unipotent element x; the smallest is a unit of order 2 with five terms in ℤS_3 = ℤGL_2(2) with π(u) = 1. They settle only the question as printed. For the restricted questions we show, with tools from Hertweck's preprint on Brauer character values whose needed parts we prove: units of order prime to p that are rationally conjugate to group elements have the property (a known argument), so a counterexample of order prime to p would contradict the first Zassenhaus conjecture; for GL_2(p), p odd, units of order p have it, and the question reduces to units of order prime to p; for GL_2(q), units of order 2 (q odd) and of odd prime order dividing q + 1 have it; and all torsion units of ℤGL_2(3) and ℤGL_2(5) have it, by published proofs of the first Zassenhaus conjecture. No printed discussion of the question was found, and the contribution is modest. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-1738-006 (Mini-Workshop: Arithmetik von Gruppenringen, Oberwolfach Report 55/2007, Problem 9 by M. Hertweck).