19 problems

Globality conjecture. Every dual pair $(M,\mathcal{L})$ of Malcev algebras and their corresponding Moufang loops is global.

Completeness conjecture. There are no other nonassociative Moufang loops of order $64$.

Let $L=(G,\circ)$ and $M=(G,*)$ be Moufang loops on the same set $G$ of order $n$, and define their distance by … Suppose that $L$ and $M$ have nonequivalent associators. Distance-…

Let $L=(G,\circ)$ and $M=(G,*)$ be Moufang $2$-loops on the same set $G$ of order $n$, and define their distance by … Quarter-distance isomorphism conjecture. If $d(\circ,*)<n^2/4$…

Let $L$ and $M$ be Moufang $2$-loops of finite order $n$, and suppose their associators are equivalent. A series of constructions similar to the cyclic and dihedral constructions i…

Let $n$ be the order of a Moufang loop, and let $\operatorname{Mlt}(L)$ denote its multiplication group. Polynomial-bound conjecture. The function … solves the problem of finding a…

Let $L$ be a nonassociative simple Moufang loop and let $H=\operatorname{Mlt}(L)_e$ be the stabilizer of the neutral element in the multiplication group of $L$. Simplicity conjectu…

Let $V \subset \mathbb{P}^3_L$ be a smooth cubic surface defined over a number field $L$. Let $\nu_1$ and $\nu_2$ be two different places of $L$. Suppose that there exists a non-as…

Let $n\geq 3$ and let $\mathbf{a}=(a_0,\ldots,a_{n-1})$ define a Fulton toric cycle, meaning that … A lifting of the Fulton cycle $A(\mathbf{a})$ is a nice cycle…

Sofic–Moufang conjecture. A group $G$ is sofic if and only if it is locally embeddable into finite Moufang loops.

Local embeddability conjecture. Every group is locally embeddable into finite Moufang loops.

Doro's conjecture. If $\operatorname{Nuc}(M)=1$, then $\operatorname{C}(M)$ is normal in $M$.

Exhaustiveness conjecture. No additional nonassociative Moufang loops of order $64$ exist; equivalently, the construction gives all nonassociative Moufang loops of order $64$.

Bruck–Paige conjecture. Every diassociative automorphic loop is Moufang.

Let $L$ be a finite Moufang $p$-loop, where $p\ne 3$. Suppose $L$ is generated by a set $\{a_i\mid i\in I\}$ satisfying … for all $i,j,k\in I$, where $(x,y,z)$ denotes the associat…

Let $E$ be a Moufang loop fitting into a short exact sequence … where $M$ is a finite simple noncyclic Moufang loop and $U$ is an abelian group. Call the extension minimal if $U$ c…

Let $Q$ and $Q'$ be automorphic Moufang loops, meaning that all inner mappings of each loop are automorphisms. A half-isomorphism is a bijective map $\varphi:Q\to Q'$ such that, fo…

Minimal identities conjecture. These identities form a minimal set of identities defining the variety ${\cal E}$.

Let $L$ be the Moufang loop constructed in the source, and let octonion algebras be the algebras whose norm-one unit loops are Moufang. Moufang-loop and octonion-algebra conjecture…