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 \(P_n(\mathbb{Z}_{p^m})\) of all \(n\times n\) matrices over \(\mathbb{Z}/p^m\) with diagonal entries \(\equiv1\) and entries below the diagonal \(\equiv0\pmod p\), a Sylow \(p\)-subgroup of \(GL_n(\mathbb{Z}/p^m)\), is regular. It is known to be regular when \(nm-1<p\) (by Merzlyakov's bound for the class) and when \(n^2<p\) (Kolesnikov), and the answer is complete for \(m\le2\) (Yagzhev for \(m=1\); Kolesnikov, and Egorychev, Kolesnikov and Leontiev, for \(m=2\)) and for \(m=3\) and \(p\equiv1\pmod3\), where Kolesnikov and Leontiev proved irregularity for \(n\ge(p+2)/3\). Our result is a partial answer: \(P_n(\mathbb{Z}/p^m)\) 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 \(P_n(\mathbb{Z}/p^3)\) is irregular for \(n=(p+4)/3\) and \(p\in\{11,17,23,29\}\); hence for every prime \(p\le29\), \(P_n(\mathbb{Z}/p^3)\) is regular iff \(3n-1\le p\), and for \(p=11\) the answer is complete except for \(n=4\), \(m\ge4\); 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 \(n^2<p\) holds, whereas for \(m\le2\) 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 \(\mho_1(\langle a,b\rangle')\) 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 \(\mathbb{F}_p\) 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 \(S\)\({}=e_{21}\)\({}+e_{32}\)\({}+\)\(\dots\)\({}+e_{n,n-1}\) and any other element of the first subdiagonal, not a multiple of \(S\), generate a Lie subalgebra of \(\mathfrak{gl}_n(\mathbb{F}_p)\) that contains \(e_{n1}\); 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 \(n\ge5\). 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.