Kourovka Notebook Problem 13.19

Definitions and mathematical statement

Let $n\geq 1$ be an integer, let $G_1,\ldots,G_n$ be groups, and write $\pi_i:G_1\times\cdots\times G_n\to G_i$ for the coordinate projections. Suppose that \[ H\trianglelefteq Q\leq G_1\times\cdots\times G_n, \qquad \pi_i(H)=\pi_i(Q)=G_i\quad(1\leq i\leq n). \] Thus both $H$ and $Q$ are subdirect subgroups of the same finite direct product. Put $P=Q/H$, and let $p$ be a prime.

A finite $p$-group is a finite group of order $p^m$ for some integer $m\geq 0$. Such a group $P$ is called regular if, for every $a,b\in P$, there are an integer $r\geq 0$ and elements $c_1,\ldots,c_r\in [K,K]$, where $K=\langle a,b\rangle$ and $[K,K]$ is its commutator subgroup, such that \[ (ab)^p=a^p b^p c_1^p\cdots c_r^p. \] The product is the identity when $r=0$.

The supplied question is: if $P=Q/H$ is a $p$-group, must it be a regular $p$-group? Under the conventional finite interpretation, this asks whether, for every prime $p$, every integer $n\geq 1$, and every choice of groups and subgroups satisfying the displayed hypotheses, a quotient $Q/H$ of finite $p$-power order satisfies the regularity condition above. No finiteness assumption on the factors, $Q$, or $H$ is stated.

The source does not explicitly require $Q/H$ to be finite or define regularity. The term $p$-group can also allow infinite groups in which every element has finite $p$-power order. Consequently, the finite formulation above is a conditional interpretation; an intended infinite-group formulation and its regularity convention remain unresolved.

Short English statement

If a group and a normal subgroup both project onto every factor of a finite direct product, must their quotient, when a $p$-group, be regular? The supplied record leaves the finiteness and regularity conventions unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $700937$, exact source problem code catalogue entry; supplied complete source record, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original Problem 13.19, 13th issue (1995); proposed by Yu. M. Gorchakov.
• The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The registry collection identifies E. I. Khukhro and V. D. Mazurov as editors of the notebook covering Issues 1--21 (1965--2026).
• The registry labels this problem open and records an update dated 2026-10-16. This is supplied source metadata, not an independently verified assessment of the mathematical or literature status.