As in other areas of mathematics, such problems are often made public at professional conferences and meetings.
Let L be a Moufang loop with normal abelian subgroup (associative subloop) M of odd order such that L/M is a cyclic group of order bigger than 3.
(ii) If the orders of M and L/M are relatively prime, is L a group?
Conjecture: Let L be a finite Moufang loop and Φ(L) the intersection of all maximal subloops of L. Then Φ(L) is a normal nilpotent subloop of L. For a group
Find a minimal presentation for the Moufang loop
If q is not congruent to 1 modulo p, are all Moufang loops of order p2q3 groups?
Is there a Moufang loop of odd order with trivial nucleus?
Conjecture: Let M be a finite Moufang loop of exponent n with m generators.
Conjecture: Let L be a finitely generated Moufang loop of exponent 4 or 6.
Let MFn be the free Moufang loop with n generators.
Call a class of finite quasigroups quadratic if there is a positive real number
Determine the Campbell–Hausdorff series for analytic Bol loops.
Is there a finite, universally flexible loop that is not middle Bol?
Is there a finite simple nonassociative Bol loop with nontrivial conjugacy classes?
Determine the number of nilpotent loops of order 24 up to isomorphism.
Construct a latin square L of order n as follows: Let G = Kn,n be the complete bipartite graph with distinct weights on its n2 edges.
Each matching Mi determines a permutation pi of 1, ..., n. Let L be obtained from G by placing the permutation pi into row i of L. Does this procedure result in a uniform distribution on the space of Latin squares of order n?
We say that a variety V of loops satisfies the Moufang theorem if for every loop Q in V the following implication holds: for every x, y, z in Q, if x(yz) = (xy)z then the subloop generated by x, y, z is a group.
A loop is Osborn if it satisfies the identity x((yz)x) = (xλ\y)(zx).
If not, is there a nice identity characterizing universal Osborn loops?
The following problems were posed as open at various conferences and have since been solved.
Is there a finite simple Buchsteiner loop that is not conjugacy closed?
Classify nonassociative Moufang loops of order 64.
Is there a finite non-Moufang left Bol loop with trivial right nucleus?
Does every finite Moufang loop have the strong Lagrange property?
Is the class of cores of Bol loops a quasivariety?
Let I(n) be the number of isomorphism classes of quasigroups of order n. Is I(n) odd for every n?
Classify the finite simple paramedial quasigroups.