Kourovka Notebook Problem 1.67

Definitions and mathematical statement

Let $G$ be a finitely presented group, and let \[ d(G)=\min\{\lvert S\rvert:S\subseteq G,\ \langle S\rangle=G\} \] be its minimal number of generators. Let $F$ be a free group of rank $d(G)$, fix a surjective homomorphism $\varphi:F\twoheadrightarrow G$, and put $N=\ker\varphi$. Define \[ [F,N]=\langle fnf^{-1}n^{-1}:f\in F,\ n\in N\rangle, \qquad Q_{\varphi}=N/[F,N]. \] Since $N$ is normal in $F$, the subgroup $[F,N]$ is normal in $N$; moreover, $[N,N]\subseteq[F,N]$, so $Q_{\varphi}$ is abelian.

Problem (L. Ya. Kulikov, 1965; Kourovka Notebook 1.67). For every such $G$, $F$, and fixed $\varphi$, find a complete system of invariants of the group $Q_{\varphi}$. Completeness means that, for any two instances satisfying these hypotheses, their invariants agree if and only if their quotient groups $Q_{\varphi}$ are isomorphic as abstract groups.

The supplied statement does not specify the required form of these invariants or whether an effective procedure for obtaining them from presentation data is required. The fixed epimorphism is retained as part of the input; no independence of its choice is assumed.

Short English statement

Given a finitely presented group $G$, map a free group with the smallest possible number of free generators onto $G$, and let $N$ be the kernel. Find invariants that completely determine the abstract isomorphism type of $N/[F,N]$. The source leaves the required form and computational nature of the invariants unspecified.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700058$, exact source problem code catalogue entry, title Kourovka Notebook Problem 1.67.
• Supplied primary-source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 1.67, first issue (1965); proposed by L. Ya. Kulikov.
• The supplied background identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description identifies E. I. Khukhro and V. D. Mazurov as editors of the Issues 1--21 collection (1965--2026). These bibliographic details are supplied attribution and have not been independently verified.
• The local registry labels the problem open. This is a supplied status label, not an independently verified assessment of the literature.