Kourovka Notebook Problem 8.3

Definitions and mathematical statement

Let $m,n\in{\mathbb{Z}}_{>0}$, let $p$ be a prime, and put $R_m={\mathbb{Z}}/p^m{\mathbb{Z}}$. Write $\rho:R_m\to{\mathbb{F}}_p$ for reduction modulo $p$, also applied entrywise to matrices. Define \[ U_n({\mathbb{F}}_p)=\{(u_{ij})\in\operatorname{GL}_n({\mathbb{F}}_p):u_{ii}=1\text{ for all }i,\ u_{ij}=0\text{ for }i>j\}. \] The supplied statement describes $P_n(R_m)$ by requiring $a_{ii}\equiv1$ for every $i$, without specifying the modulus, and $a_{ij}\equiv0\pmod p$ for $i>j$. If the diagonal congruence is also modulo $p$, this gives the finite $p$-group \[ P_n(R_m)=\{A=(a_{ij})\in\operatorname{GL}_n(R_m):\rho(A)\in U_n({\mathbb{F}}_p)\}. \] There is no further restriction on entries above the diagonal. This interpretation of the diagonal condition requires confirmation: imposing $a_{ii}=1$ in $R_m$ instead does not in general give a group under matrix multiplication. The formulation below is conditional on the modulo-$p$ interpretation and does not silently resolve the source omission.

For a group $H$, write $H'=[H,H]$ for its commutator subgroup. A finite $p$-group $G$ is called regular if, for every $x,y\in G$, there exist an integer $r\geq0$ and elements $c_1,\ldots,c_r\in\langle x,y\rangle'$ such that \[ (xy)^p=x^py^p c_1^p\cdots c_r^p, \] where the product is the identity when $r=0$.

Problem (B. A. F. Wehrfritz, 1982). For every prime $p$, determine exactly which pairs $(m,n)\in{\mathbb{Z}}_{>0}^2$ make $P_n({\mathbb{Z}}/p^m{\mathbb{Z}})$ regular, subject to confirmation of the diagonal congruence above.

Supplied editorial qualifications. The editors' comment dated 2001 reports that regularity holds when $mn<p$, citing Yu. I. Merzlyakov [3]. It reports that A. V. Yagzhev [4] completely treated $m=1$, while explicitly warning that that paper is erroneous for $m>1$. It also reports that S. G. Kolesnikov [5] completed the case $m=2$. The editors' comment dated 2009 reports that Kolesnikov [6] proved regularity for every positive integer $m$ whenever $n^2<p$. These are supplied historical assessments; the complete classifications for $m=1$ and $m=2$ are not stated in the record. The local registry labels the problem open; that label is not an independently verified literature assessment.

Short English statement

For each prime $p$, classify the positive integers $m,n$ for which the indicated matrix group over ${\mathbb{Z}}/p^m{\mathbb{Z}}$ is a regular finite $p$-group. The source omits the modulus in its diagonal condition, so this draft conditionally reads that condition modulo $p$. Preserve the supplied results and warnings for special cases; the general open label has not been independently verified.

Sources

• {[1 }] Local UnsolvedMath registry, supplied complete record with permanent ID $700455$, exact source problem code catalogue entry, title Kourovka Notebook Problem 8.3. Supplied status: open. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82; record updated 2026-10-16T14:00:00Z.
• {[2 }] The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 8.3, proposed by B. A. F. Wehrfritz in the 8th issue (1982). The supplied record identifies the 21st-issue edition (2026), arXiv:1401.0300v48. The collection description credits editors E. I. Khukhro and V. D. Mazurov. Editorial comments dated 2001 and 2009 are reproduced as supplied historical assessments, not independently verified findings.
• {[3 }] Yu. I. Merzlyakov, Algebra i Logika 3, no. 4 (1964), 49--59 (Russian). The supplied 2001 editorial comment attributes regularity under $mn<p$ to this reference; no article title is supplied.
• {[4 }] A. V. Yagzhev, Math. Notes 56, nos. 5--6 (1995), 1283--1290. The supplied 2001 editorial comment says that this paper completely treats $m=1$, but is erroneous for $m>1$; no article title is supplied.
• {[5 }] S. G. Kolesnikov, Issledovaniya in analysis and algebra, no. 3, Tomsk Univ., Tomsk (2001), 117--125 (Russian), with the publication name reproduced as supplied. The supplied 2001 editorial comment attributes completion of $m=2$ to this reference; no article title is supplied.
• {[6 }] S. G. Kolesnikov, Siberian Math. J. 47, no. 6 (2006), 1054--1059. The supplied 2009 editorial comment attributes regularity for every $m$ when $n^2<p$ to this reference; no article title is supplied.