Kourovka Notebook Problem 12.58

Definitions and mathematical statement

For the parameter pairs $(s,t)=(4,11)$ and $(s,t)=(5,7)$, consider an incidence system $\mathcal G=(P,\mathcal L,I)$, where $P$ is a set of points, $\mathcal L$ is a set of lines, and $I\subseteq P\times\mathcal L$ is the incidence relation. Write \[ P_L=\{p\in P:(p,L)\in I\},\qquad \mathcal L_p=\{L\in\mathcal L:(p,L)\in I\}. \] The supplied definition of a generalized quadrangle $GQ(s,t)$ requires \[ \begin{aligned} &|P_L|=s+1 &&\text{for every }L\in\mathcal L,\\ &|P_L\cap P_M|\leq 1 &&\text{for all distinct }L,M\in\mathcal L,\\ &|\mathcal L_p|=t+1 &&\text{for every }p\in P, \end{aligned} \] and, for every $p\in P$ and $L\in\mathcal L$ with $p\notin P_L$, \[ \left|\{M\in\mathcal L_p:P_M\cap P_L\neq\varnothing\}\right|=1. \] For the intended questions below, take $P\neq\varnothing$. This nonemptiness requirement makes explicit the conventional intended reading; the supplied definition does not state it, and its listed axioms alone permit an empty incidence system. The record also does not explicitly impose finiteness on $P$ or $\mathcal L$; no separate finiteness hypothesis is added here.

An incidence-preserving automorphism of $\mathcal G$ is a pair of bijections $\sigma_P:P\to P$ and $\sigma_{\mathcal L}:\mathcal L\to\mathcal L$ such that \[ (p,L)\in I\quad\Longleftrightarrow\quad (\sigma_P(p),\sigma_{\mathcal L}(L))\in I. \] Write $\operatorname{Aut}(\mathcal G)$ for this group. An involution is an element $\sigma\in\operatorname{Aut}(\mathcal G)$ satisfying $\sigma^2=1$ and $\sigma\neq 1$.

1. (a) Does there exist a nonempty generalized quadrangle $GQ(4,11)$ satisfying the stated axioms?
2. (b) For a hypothetical nonempty generalized quadrangle $\mathcal G$ with parameters $(5,7)$, can $\operatorname{Aut}(\mathcal G)$ contain an involution? Equivalently, can there exist such a pair $(\mathcal G,\sigma)$ with $\sigma\in\operatorname{Aut}(\mathcal G)$, $\sigma\neq 1$, and $\sigma^2=1$? This question does not assume or assert the existence of a $GQ(5,7)$.

Here automorphisms preserve points and lines separately, as in the usual incidence-automorphism convention; the supplied record does not explicitly discuss incidence symmetries interchanging these types.

Short English statement

The questions ask whether a generalized quadrangle with parameters $(4,11)$ exists and whether a hypothetical generalized quadrangle with parameters $(5,7)$ can have an automorphism of order two. Nonemptiness and the convention that automorphisms preserve points and lines separately are made explicit and require source review.

Sources

• Local UnsolvedMath registry, permanent ID $700877$, exact source problem code catalogue entry, title Kourovka Notebook Problem 12.58. The supplied record labels the problem open; this is a supplied registry classification, not an independently verified mathematical or literature assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem $12.58$, 12th issue (1992), attributed to A. A. Makhnëv. The supplied background identifies the 21st issue edition (2026), arXiv:1401.0300v48. This bibliographic attribution is supplied evidence and has not been independently checked.
• The supplied collection description identifies E. I. Khukhro and V. D. Mazurov as editors of the Kourovka Notebook collection covering Issues 1--21 (1965--2026). The registry record was updated on 2026-10-16; this metadata date is not a source-verification date. No separate dated literature assessment is supplied.