Kourovka Notebook Problem 18.90

Definitions and mathematical statement

The supplied record does not explicitly specify the allowed values of the parameters $m,n$. The formulation below uses positive integers; this intended parameter range requires confirmation.

For $m,n\in{\mathbb{Z}}_{\geq 1}$, let \[ Y(m,n)=\left\langle a_1,\ldots,a_m\ \middle|\ a_k^n=1\ (1\leq k\leq m),\quad (a_k^i a_l^i)^2=1\ \left(1\leq k<l\leq m,\ 1\leq i\leq\left\lfloor\frac n2\right\rfloor\right)\right\rangle. \] All indices in this presentation are integers, and $1$ denotes the group identity. An empty range contributes no relations. Subject to confirmation of the parameter range, the requested assertion is \[ \forall m,n\in{\mathbb{Z}}_{\geq 1},\qquad |Y(m,n)|<\infty. \]

The following are contextual claims supplied with the problem, not independently verified assertions of this draft. Write $C_n$ for the cyclic group of order $n$, $A_r$ for the alternating group of degree $r$, and $C_2^m$ for the direct product of $m$ copies of $C_2$. The source states \[ Y(m,2)\cong C_2^m,\qquad Y(m,3)\cong A_{m+2}, \] attributes the latter presentation to Carmichael, and states that $Y(m,4)$ has order $2^{m(m+3)/2}$, nilpotency class $3$, and exponent $4$. The class claim needs a qualification on $m$: at $m=1$, the defining presentation gives the cyclic group $C_4$, which has nilpotency class $1$.

Let ${\mathbb{F}}_2[C_n]$ be the group algebra and let its augmentation ideal be \[ \omega(C_n)=\ker\!\left({\mathbb{F}}_2[C_n]\longrightarrow{\mathbb{F}}_2,\quad \sum_{g\in C_n}c_g g\longmapsto\sum_{g\in C_n}c_g\right). \] The source describes $Y(2,n)$ as the natural extension of $\omega(C_n)$ by $C_n$, attributing its presentation to Coxeter, and states $Y(3,n)\cong SL(2,\omega(C_n))$. The extension data and the meaning of $SL(2,\omega(C_n))$, whose ring argument is generally nonunital, are not specified.

For odd $n$, the source additionally says that $Y(m,n)$ has the following presentation, but labels the presented group with a lowercase $y$: \[ \begin{aligned} y(m,n)=\big\langle a,\mathbb S_m\ \big|\;&a^n=1,\\ &[\tau_{12}^{a^i},\tau_{12}]=1\quad\left(1\leq i\leq\left\lfloor\frac n2\right\rfloor\right),\\ &\tau_{12}^{1+a+\cdots+a^{n-1}}=1,\\ &\tau_{i,i+1}a\tau_{i,i+1}=a^{-1}\quad(2\leq i\leq m-1)\big\rangle. \end{aligned} \] Here $\mathbb S_m$ is the symmetric group on $m$ letters, with its own group relations understood, and $\tau_{ij}$ denotes the transposition interchanging $i$ and $j$. This display requires $m\geq2$. The source does not explain the relationship between its uppercase and lowercase group names, its conjugation convention, or the word represented by the formal-sum exponent $1+a+\cdots+a^{n-1}$. Consequently this auxiliary presentation is retained as supplied context and is not used as a verified equivalent definition of $Y(m,n)$.

The source reports the following examples, with its classical-group notation retained: \[ \begin{aligned} y(3,5)&\cong SL(2,16)\cong\Omega^{-}(4,4),\\ y(4,5)&\cong Sp(2,16)\cong\Omega(5,4),\\ y(5,5)&\cong SU(4,16)\cong\Omega^{-}(6,4),\\ y(6,5)&\cong 4^6\Omega^{-}(6,4),\\ y(3,7)&\cong SL(2,8)^2\cong\Omega^{+}(4,8),\\ y(4,7)&\cong Sp(4,8)\cong\Omega(5,8). \end{aligned} \] The classical-group parameter conventions and the extension notation $4^6\Omega^{-}(6,4)$ require clarification before these examples can be used precisely. The source describes connections with classical groups in characteristic $2$, attributes extensive computations confirming this trend to Felsch, Neubüser and O'Brien, and mentions Bott periodicity and Clifford algebras without giving a precise additional assertion.

Short English statement

S. Sidki's problem asks whether every group $Y(m,n)$ defined by the displayed generators and relations is finite. The intended range of $m,n$ is unstated; positive integers are used provisionally. The supplied registry labels the problem open. Its auxiliary presentation and contextual examples contain unresolved notation and qualifications and require human mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701519$, exact source problem code catalogue entry, title Kourovka Notebook Problem 18.90. The record labels the problem open; this is supplied registry status, not an independently verified literature assessment. Its recorded update date is 2026-10-16T14:00:00Z, and its upstream revision is d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied primary-source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 18.90, proposed by S. Sidki in the 18th issue (2014). The background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are reproduced from the record; the cited edition has not been consulted for this draft.
• Contextual attributions supplied in the statement: Carmichael for the presentation associated with $Y(m,3)$; Coxeter for the presentation associated with $Y(2,n)$; and Felsch, Neubüser and O'Brien for extensive computations. No separate bibliographic citations or links for these contributions are supplied.