Kourovka Notebook Problem 15.88

Definitions and mathematical statement

Let $\mathfrak A$ be the variety of abelian groups and $\mathfrak N_c$ the variety of nilpotent groups of class at most $c$. Under the standard convention for products of group varieties, $G\in\mathfrak A\mathfrak N_c$ means that there exists an abelian normal subgroup $A\trianglelefteq G$ such that $G/A\in\mathfrak N_c$.

Let $F_r=F_r(\mathfrak A\mathfrak N_c)$ be the relatively free group of rank $r$ in this variety. Thus a free generating set $X$ of cardinality $r$ has the property that every map from $X$ to a group in $\mathfrak A\mathfrak N_c$ extends uniquely to a homomorphism from $F_r$. An element $g\in F_r$ is primitive if it belongs to some free basis of $F_r$.

Question. Does there exist an algorithm recognizing primitive elements of $F_r$? Under the usual interpretation with fixed positive integers $c$ and $r$, fix a free basis $X=\{x_1,\ldots,x_r\}$. The requested algorithm must halt on every finite word $w$ in $X^{\pm1}$ and decide whether its value $g=w(X)\in F_r$ satisfies \[ \exists\,(b_1,\ldots,b_r)\text{ a free basis of }F_r\quad b_1=g. \] The supplied source does not explicitly specify the ranges of $c$ and $r$, whether infinite rank or the boundary case $c=0$ is included, or whether a single algorithm uniform in $c$ and $r$ is required. The finite-rank, fixed-parameter formulation above is therefore an explicit interpretation requiring review.

Short English statement

The problem asks whether one can decide, from a word in a relatively free abelian-by-nilpotent group, whether the represented element belongs to a free basis. The intended parameter ranges and any requirement of uniformity remain unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $701176$, exact source problem code catalogue entry, titled Kourovka Notebook Problem 15.88. The supplied record labels the problem open; this status has not been independently verified.
• Attributed in the supplied record to E. I. Timoshenko, proposed in 2002. The stated source is The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), Problem 15.88, as carried in the supplied 21st-issue edition reference (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution and have not been independently checked.