Kourovka Notebook Problem 18.120

Definitions and mathematical statement

Let $p$ be a prime, and let $P$ be a finite group of order $p^n$ for some integer $n\geq 0$. Let $A,B\leq P$ satisfy \[ P=AB=\{ab:a\in A,\ b\in B\}, \] where $A$ is abelian and $B$ is nilpotent of class two. Write \[ [B,B]=\langle x^{-1}y^{-1}xy:x,y\in B\rangle, \qquad Z(B)=\{z\in B:zb=bz\text{ for every }b\in B\}. \] The class-two condition gives $[B,B]\leq Z(B)$. If "class two" means class exactly two, additionally require $[B,B]\neq\{1\}$; the supplied record does not specify whether class at most two is permitted.

The question is whether, for every such choice of $p,P,A,B$, there exists a subgroup $B_0\leq B$ such that \[ [B_0,B_0]=\{1\} \qquad\text{and}\qquad \langle A,[B,B]\rangle=AB_0, \] where $\langle A,[B,B]\rangle$ is the subgroup generated by $A$ and $[B,B]$, and $AB_0=\{ab_0:a\in A,\ b_0\in B_0\}$ denotes a set product.

The source adds "Suppose, if necessary, $A\cap B=1$." Thus it also permits considering the same universal assertion restricted to factorizations satisfying $A\cap B=\{1\}$. This is a distinct, weaker formulation; the supplied wording does not fix whether the intended target requires the unrestricted assertion or allows the restricted assertion.

Short English statement

For a finite $p$-group that is the product of an abelian subgroup $A$ and a class-two subgroup $B$, must the subgroup generated by $A$ and the commutator subgroup of $B$ equal $AB_0$ for some abelian subgroup $B_0\leq B$? The source permits adding $A\cap B=\{1\}$ if necessary, leaving the intended scope unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $701548$, source code catalogue entry, title Kourovka Notebook Problem 18.120; supplied complete source record.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 18.120, 18th issue (2014), proposed by E. Jabara. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels this problem open. This is a supplied status label, not an independently verified mathematical or literature assessment. The cited source has not been independently consulted for this draft.