22 problems

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…

Kearnes' conjecture. Every F-quasigroup is isotopic to a Moufang loop.

Non-subloop conjecture. There exists a PACC-loop $Q$ in which $M(Q)$ is not a subloop.

Lower-bound conjecture. The order of $C$ satisfies

Special half-isomorphism conjecture. Every half-isomorphism between automorphic loops is special.

Let $B(X)$ be a free Bol loop of exponent two. An automorphism of $B(X)$ is tame if it belongs to the group generated by the elementary automorphisms that, for a free generating se…

A free Bol loop of exponent two is a Bol loop freely generated in the variety of Bol loops satisfying $x^2=1$. Subloop freeness conjecture. Every subloop of a free Bol loop of expo…

Let $G$ be a uniquely 2-divisible group, and let $(G,\circ)$ be the commutative loop arising from the construction in which $G$ is equipped with the operation $\circ$. A group is m…

Let $p$ be a prime, let $M_2(\mathbb F_p)$ denote the $2\times 2$ matrices over $\mathbb F_p$, and let $Q_{\mathbb F_p}(A)$ be the loop associated with a matrix…

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

Let $Q$ be a loop, and let $\operatorname{Inn}(Q)$ denote its inner mapping group. A loop is centrally nilpotent if it has a central series; its nilpotence class is the length of t…

Group-table modification conjecture. There exists a group $G$ and a fixed central involution $h\in G$ such that $C=(G,*)$ and, for all $x,y\in G$,

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

The Main Conjecture. The quotient $Q/N(Q)$ is an abelian group and $Q/Z(Q)$ is a group. In particular, $Q$ is nilpotent of class at most $3$.

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 $Q$ 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…

Degree-and-type conjecture. If $V_{1}$ and $V_{2}$ have the same degree and type, then they are isomorphic.

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

Isomorphism-type conjecture. The loops

Classification conjecture. The class of loops constructed below accounts for all commutative A-loops of order $p^3$.