Kourovka Notebook Problem 16.46

Definitions and mathematical statement

For a group presentation, every listed relator is set equal to the identity. Write $[x,y]=x^{-1}y^{-1}xy$. Define \[ \begin{aligned} G_3=\langle h,a,P,Q\mid {}&h^3,a^2,P^2,Q^2,[P,Q],[h,P],(hQ)^2,a^{-1}PaQ\rangle,\\ G_4=\langle h,a,p,q,r\mid {}&h^3,a^2,p^2,q^2,r^2,[p,q],[p,r],p(qr)^2,\\ &h^{-1}phq,h^{-1}qhpq,(hr)^2,[a,p],a^{-1}qar\rangle. \end{aligned} \] The supplied record states that both groups act arc-transitively on the infinite $3$-valent tree with finite vertex stabilizers. Here every vertex of the tree has degree $3$, and arc-transitivity means transitivity on ordered pairs of adjacent vertices. It also states that each group contains \[ G_1=\langle h,a\mid h^3,a^2\rangle\cong\operatorname{PSL}_2({\mathbb{Z}}) \] as a subgroup of finite index.

Form the amalgamated free product $G=G_3*_{G_1}G_4$, identifying the displayed copies of $h$ and $a$. According to the supplied record, $G$ has a normal subgroup $K$ of index $8$, generated by \[ A=h,\qquad B=aha,\qquad C=p,\qquad D=PpP,\qquad E=QpQ,\qquad F=PQpPQ, \] with dihedral complement $\langle a,P,Q\rangle$; thus the asserted decomposition is $G=K\rtimes\langle a,P,Q\rangle$, with this complement of order $8$. The supplied presentation of $K$, which specifies the group asked about, is \[ \begin{aligned} K=\langle A,B,C,D,E,F\mid {}&A^3,B^3,C^2,D^2,E^2,F^2,\\ &(AC)^3,(AD)^3,(AE)^3,(AF)^3,\\ &(BC)^3,(BD)^3,(BE)^3,(BF)^3,\\ &(ABA^{-1}C)^2,(ABA^{-1}D)^2,\\ &(A^{-1}BAE)^2,(A^{-1}BAF)^2,\\ &(BAB^{-1}C)^2,(B^{-1}ABD)^2,\\ &(BAB^{-1}E)^2,(B^{-1}ABF)^2\rangle. \end{aligned} \] Question. Does there exist a finite group $H$ with $|H|>1$ and a surjective homomorphism $K\twoheadrightarrow H$? Equivalently, does there exist $N\trianglelefteq K$ such that $1<[K:N]<\infty$? The structural assertions preceding the question are supplied source assertions and have not been independently verified here.

Short English statement

Does the explicitly presented group $K$, described in the supplied record as a normal subgroup of index $8$ in $G_3*_{G_1}G_4$, have a nontrivial finite quotient? The registry labels the problem open; that label is supplied metadata, not an independently verified assessment.

Sources

• Local UnsolvedMath registry, permanent ID $701238$, exact source code catalogue entry, title Kourovka Notebook Problem 16.46. The supplied record gives upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82 and an update timestamp of 2026-10-16T14:00:00Z.
• M. Conder, Problem 16.46, The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), as attributed in the supplied record. The record also identifies the 21st issue edition (2026), arXiv:1401.0300v48. Its collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution; the cited edition has not been independently consulted here.