Extending Generator Maps to Finite Group Homomorphisms

Definitions and mathematical statement

The explicit statement in the supplied registry concerns integral matrix groups. Let $n\geq 1$, let $g_1,\ldots,g_k$ be a finite list of matrices in \[ \operatorname{SL}(n,{\mathbb{Z}})=\{A\in M_n({\mathbb{Z}}):\det A=1\}, \] and put $G=\langle g_1,\ldots,g_k\rangle$. Let $H$ be a finite group with specified generators $h_1,\ldots,h_k$, and consider the prescribed assignment $g_i\mapsto h_i$.

The requested determination is whether there exists a group homomorphism \[ \varphi:G\longrightarrow H \quad\text{such that}\quad \varphi(g_i)=h_i\quad\text{for every }i\in\{1,\ldots,k\}. \] Any such extension is necessarily surjective because the prescribed images generate $H$.

For an integer $m\geq 2$, let \[ \rho_m:G\longrightarrow\operatorname{SL}(n,{\mathbb{Z}}/m{\mathbb{Z}}) \] be reduction of matrix entries modulo $m$. In the standard interpretation used here, an extension factors through a congruence homomorphism if, for some $m\geq2$, there is a homomorphism $\psi_m:\rho_m(G)\to H$ satisfying $\varphi=\psi_m\circ\rho_m$. The requested determination must include potential extensions for which no such factorization exists for any $m\geq2$; it is not restricted to congruence quotients.

There is an unresolved discrepancy in the supplied record: its literature assessment refers to a permutation group and a small finite target group, whereas its original, clean, and displayed statements specify $G\leq\operatorname{SL}(n,{\mathbb{Z}})$ and an arbitrary finite group $H$. The formulation above follows those explicit statements and does not resolve that discrepancy. The record also does not specify an encoding of $H$, a required complexity bound, or a precise algorithmic success criterion beyond determining whether the extension exists.

Short English statement

Given generators of an integral matrix group and prescribed images generating a finite group, determine whether those images extend to a group homomorphism, including extensions that do not factor through reduction modulo any integer. The supplied background describes a different input setting, so the intended computational scope requires review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004770$, title Extending Generator Maps to Finite Group Homomorphisms, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. The supplied proposed year is 2021; no individual proposer is identified.
• Supplied source citation: Computational Group Theory (2022), Oberwolfach Reports, pp. 2027--2087, DOI 10.4171/owr/2021/38. The source URL and the literature-source link labelled Oberwolfach Report 38/2021 are https://doi.org/10.4171/owr/2021/38. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• The supplied literature assessment was checked on 2026-08-22 and labels the problem open. It describes extension of an arbitrary generator assignment from a permutation group to a small finite group, especially outside congruence quotients, as an unresolved verification problem. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry supplies the statement-status label corrected_verified and reports that the official report was checked directly and that the clean statement restores definitions and removes extraction artifacts while preserving the mathematical target. These are supplied verification claims, not verification performed for this draft.