Regularity of the Sylow p-Subgroups of GL_n(Z/p^m) at the Class-p Boundary: New Cases of a Question of Wehrfritz

In Problem 8.3 of the Kourovka Notebook (1982), B. A. F. Wehrfritz asked for which m and n the group P_n(Zp^m) of all n×n matrices over Z/p^m with diagonal entries ≡ 1 and entries below the diagonal ≡ 0 (mod p), a Sylow p-subgroup of GL_n(Z/p^m), is regular. It is known to be regular when nm − 1 < p (by Merzlyakov's bound for the class) and when n² < p (Kolesnikov), and the answer is complete for m ≤ 2 (Yagzhev for m = 1; Kolesnikov, and Egorychev, Kolesnikov and Leontiev, for m = 2) and for m = 3 and p ≡ 1 (mod 3), where Kolesnikov and Leontiev proved irregularity for n ≥ (p+2)/3. Our result is a partial answer: P_n(Z/p^m) is regular in the boundary cases nm − 1 = p for (p, n, m) = (11,4,3), (17,6,3), (23,8,3), (29,10,3), (19,5,4), (23,6,4), (29,6,5), and P_n(Z/p^3) is irregular for n = (p+4)/3 and p ∈ {11, 17, 23, 29}; hence for every prime p ≤ 29, P_n(Z/p^3) is regular iff 3n − 1 ≤ p, and for p = 11 the answer is complete except for n = 4, m ≥ 4; the question stays open in general. In particular, for m = 3, 4, 5 the group can be regular although neither of the known sufficient conditions nm − 1 < p and n² < p holds, whereas for m ≤ 2 the condition nm − 1 < p is necessary. All new cases rest on computer-assisted proofs. The irregularity witnesses are the pair of matrices used by Kolesnikov and Leontiev, for which we compute the agemo subgroup ℧_1(⟨a,b⟩′) exactly. For regularity, the Hall–Petresco identity and the Magnus embedding reduce the question, at the boundary nm − 1 = p, to a finite computation in linear algebra over F_p on the two-dimensional subspaces of an n-dimensional space, shortened by a torus symmetry. For m = 3 we prove by hand that the boundary case n = (p+1)/3 is regular whenever S = e_21 + e_32 + ⋯ + en,n−1 and any other element of the first subdiagonal, not a multiple of S, generate a Lie subalgebra of gl_n(F_p) that contains eₙ₁; we verify this condition by computer for n = 4, 6, 8, 10, which gives the new cases (23,8,3) and (29,10,3), and show that it fails for every odd n ≥ 5. Three conjectures suggested by the computations are stated. No printed decision of the new cases was found in the sources searched (zbMATH, arXiv, Crossref, OpenAlex, Semantic Scholar, web searches); a search that finds nothing is not a proof of novelty. 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: KOU-8.3 (Kourovka Notebook, Problem 8.3, B. A. F. Wehrfritz, 8th issue, 1982).