Kourovka Notebook Problem 20.8

Definitions and mathematical statement

A free group has finite rank if it has a finite free basis; its rank is the cardinality of such a basis. For a possibly infinitely generated free subgroup $L$, let $\operatorname{rank}(L)$ denote the cardinality of a free basis. In particular, an infinite rank is greater than every finite rank.

Let $K\leq H\leq F$ be free groups of finite rank satisfying \[ \operatorname{rank}(H)<\operatorname{rank}(K) \] and \[ \forall L\leq H,\qquad \left(K\leq L\ \text{and}\ L\ne H\right) \Longrightarrow \operatorname{rank}(L)\geq\operatorname{rank}(K). \] The quantifier over $L$ includes all intermediate subgroups, whether or not they are finitely generated, and allows $L=K$.

Let $\iota:H\hookrightarrow F$ be the inclusion homomorphism. Must \[ \forall\varphi\in\operatorname{Hom}(H,F),\qquad \left(\forall k\in K,\ \varphi(k)=k\right) \Longrightarrow \varphi=\iota \] hold? Here $\varphi$ is an arbitrary group homomorphism, with no injectivity or surjectivity assumption.

The source writes $K<H<F$ without defining the symbol $<$. If it denotes proper subgroup inclusion, impose the additional hypothesis $H\ne F$; the rank inequality already implies $K\ne H$. If it denotes subgroup inclusion allowing equality, no additional hypothesis is imposed. This convention remains to be checked against the cited source.

Short English statement

Suppose a finite-rank free subgroup $K$ lies in a free group $H$ of smaller finite rank, and every proper subgroup of $H$ containing $K$ has rank at least that of $K$. For a finite-rank free ambient group $F$ containing $H$, must every homomorphism $H\to F$ that fixes $K$ pointwise be the inclusion? Whether the source requires $H$ to be proper in $F$ remains a notation qualification.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701666$, exact source problem code catalogue entry, title Kourovka Notebook Problem 20.8.
• The supplied record attributes the problem to G. M. Bergman, with proposed year 2022, and cites The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), original problem number 20.8, and the 21st issue edition (2026), arXiv:1401.0300v48. Its collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels the problem open. This is supplied source metadata, not an independently verified mathematical or literature assessment. The cited editions and attribution have not been independently checked for this draft.