Kourovka Notebook Problem 3.5

Definitions and mathematical statement

A variety of groups $\mathcal V$ is the class of all groups satisfying a specified set of group identities. A group $F$ is relatively free in $\mathcal V$ on a set $X$ if $F\in\mathcal V$ has a generating map $\iota:X\to F$ such that every map from $X$ to a group $G\in\mathcal V$ extends uniquely to a homomorphism $F\to G$.

A group $H$ is radicable if every element has an $n$-th root in $H$ for every positive integer $n$: \[ \forall h\in H\;\forall n\in{\mathbb{Z}}_{\geq 1}\;\exists u\in H:\quad u^n=h. \] In particular, roots must lie in $H$, rather than merely in an ambient group.

For a set $W$ of group words, the associated verbal subgroup of $F$ is \[ W(F)=\left\langle w(g_1,\ldots,g_{k(w)}):w\in W,\ g_1,\ldots,g_{k(w)}\in F\right\rangle, \] where $k(w)$ is the number of variables of the word $w$.

Question (N. R. Brumberg, Kourovka Notebook Problem 3.5). Can a subgroup of a relatively free group be radicable? In particular, can a verbal subgroup of a relatively free group be radicable? With unrestricted existential quantifiers, these ask whether there exist a variety $\mathcal V$, a set $X$, and a relatively free group $F$ in $\mathcal V$ on $X$ satisfying, respectively, \[ \exists H\leq F\;\forall h\in H\;\forall n\in{\mathbb{Z}}_{\geq1}\;\exists u\in H:\quad u^n=h, \] or \[ \exists W\;\forall h\in W(F)\;\forall n\in{\mathbb{Z}}_{\geq1}\;\exists u\in W(F):\quad u^n=h. \] Unresolved qualification. The supplied wording does not require the subgroup to be nontrivial. The trivial group satisfies the definition of radicability, so the intended substantive question requires clarification about whether $H\ne\{1\}$, respectively $W(F)\ne\{1\}$, is required. These restrictions have not been silently added. The source also specifies no variety, rank, finiteness condition, or properness requirement; the unrestricted quantifiers above record its literal scope, subject to this unresolved qualification.

Short English statement

Brumberg asks whether a relatively free group can contain a radicable subgroup, and more specifically a radicable verbal subgroup. Radicability requires every positive integer root to exist within the subgroup. The supplied statement leaves nontriviality unstated, so its intended substantive formulation needs human review.

Sources

• Local UnsolvedMath registry, permanent ID $700159$, source problem code catalogue entry, title Kourovka Notebook Problem 3.5; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The supplied record attributes Problem 3.5 to N. R. Brumberg, proposed in 1969, and identifies its source as The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), as included in the 21st issue edition (2026), arXiv:1401.0300v48.
• The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors of the Kourovka Notebook collection covering Issues 1--21 (1965--2026).
• The local record labels the problem open and records an update timestamp of 2026-10-16T14:00:00Z. These are supplied registry metadata, not an independently verified assessment of the literature. The cited edition has not been independently consulted for this draft.