List of problems in loop theory and quasigroup theory

In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Many of the problems posed here first appeared in the Loops (Prague) conferences and the Mile High (Denver) conferences.

Open problems (Moufang loops)

Abelian by cyclic groups resulting in Moufang loops

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. (i) Is L a group? (ii) If the orders of M and L/M are relatively prime, is L a group?

Embedding CMLs of period 3 into alternative algebras

Conjecture: Any finite commutative Moufang loop of period 3 can be embedded into a commutative alternative algebra.

Frattini subloop for Moufang loops

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.

Minimal presentations for loops M(G,2)

For a group , define on x by , , , . Find a minimal presentation for the Moufang loop with respect to a presentation for .

Moufang loops of order p2q3 and pq4

Let p and q be distinct odd primes. If q is not congruent to 1 modulo p, are all Moufang loops of order p2q3 groups? What about pq4?

(Phillips' problem) Odd order Moufang loop with trivial nucleus

Is there a Moufang loop of odd order with trivial nucleus?

Presentations for finite simple Moufang loops

Find presentations for all nonassociative finite simple Moufang loops in the variety of Moufang loops.

The restricted Burnside problem for Moufang loops

Conjecture: Let M be a finite Moufang loop of exponent n with m generators. Then there exists a function f(n,m) such that |M| < f(n,m).

The Sanov and M. Hall theorems for Moufang loops

Conjecture: Let L be a finitely generated Moufang loop of exponent 4 or 6. Then L is finite.

Torsion in free Moufang loops

Let MFn be the free Moufang loop with n generators.

Conjecture: MF3 is torsion free but MFn with n > 4 is not.

Open problems (Bol loops)

Nilpotency degree of the left multiplication group of a left Bol loop

For a left Bol loop Q, find some relation between the nilpotency degree of the left multiplication group of Q and the structure of Q.

Are two Bol loops with similar multiplication tables isomorphic?

Let , be two quasigroups defined on the same underlying set . The distance is the number of pairs in such that . Call a class of finite quasigroups quadratic if there is a positive real number such that any two quasigroups , of order from the class satisfying are isomorphic. Are Moufang loops quadratic? Are Bol loops quadratic?

Campbell–Hausdorff series for analytic Bol loops

Determine the Campbell–Hausdorff series for analytic Bol loops.

Universally flexible loop that is not middle Bol

A loop is universally flexible if every one of its loop isotopes is flexible, that is, satisfies (xy)x = x(yx). A loop is middle Bol if every one of its loop isotopes has the antiautomorphic inverse property, that is, satisfies (xy)1 = y1x1. Is there a finite, universally flexible loop that is not middle Bol?

Finite simple Bol loop with nontrivial conjugacy classes

Is there a finite simple nonassociative Bol loop with nontrivial conjugacy classes?

Open problems (Nilpotency and solvability)

Niemenmaa's conjecture and related problems

Let Q be a loop whose inner mapping group is nilpotent. Is Q nilpotent? Is Q solvable?

Loops with abelian inner mapping group

Let Q be a loop with abelian inner mapping group. Is Q nilpotent? If so, is there a bound on the nilpotency class of Q? In particular, can the nilpotency class of Q be higher than 3?

Number of nilpotent loops up to isomorphism

Determine the number of nilpotent loops of order 24 up to isomorphism.

Open problems (quasigroups)

Classification of finite simple paramedial quasigroups

Classify the finite simple paramedial quasigroups.

Existence of infinite simple paramedial quasigroups

Are there infinite simple paramedial quasigroups?

Minimal isotopically universal varieties of quasigroups

A variety V of quasigroups is isotopically universal if every quasigroup is isotopic to a member of V. Is the variety of loops a minimal isotopically universal variety? Does every isotopically universal variety contain the variety of loops or its parastrophes?

Small quasigroups with quasigroup core

Does there exist a quasigroup Q of order q = 14, 18, 26 or 42 such that the operation * defined on Q by x * y = y  xy is a quasigroup operation?

Uniform construction of Latin squares?

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. Let M1 be the cheapest matching in G, M2 the cheapest matching in G with M1 removed, and so on. 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?

Open problems (miscellaneous)

Bound on the size of multiplication groups

For a loop Q, let Mlt(Q) denote the multiplication group of Q, that is, the group generated by all left and right translations. Is |Mlt(Q)| < f(|Q|) for some variety of loops and for some polynomial f?

Does every finite alternative loop have 2-sided inverses?

Does every finite alternative loop, that is, every loop satisfying x(xy) = (xx)y and x(yy) = (xy)y, have 2-sided inverses?

Finite simple nonassociative automorphic loop

Find a nonassociative finite simple automorphic loop, if such a loop exists.

Moufang theorem in non-Moufang loops

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. Is every variety that satisfies Moufang theorem contained in the variety of Moufang loops?

Universality of Osborn loops

A loop is Osborn if it satisfies the identity x((yz)x) = (xλ\y)(zx). Is every Osborn loop universal, that is, is every isotope of an Osborn loop Osborn? If not, is there a nice identity characterizing universal Osborn loops?

Solved problems

The following problems were posed as open at various conferences and have since been solved.

Buchsteiner loop that is not conjugacy closed

Is there a Buchsteiner loop that is not conjugacy closed? Is there a finite simple Buchsteiner loop that is not conjugacy closed?

Classification of Moufang loops of order 64

Classify nonassociative Moufang loops of order 64.

Conjugacy closed loop with nonisomorphic one-sided multiplication groups

Construct a conjugacy closed loop whose left multiplication group is not isomorphic to its right multiplication group.

Existence of a finite simple Bol loop

Is there a finite simple Bol loop that is not Moufang?

Left Bol loop with trivial right nucleus

Is there a finite non-Moufang left Bol loop with trivial right nucleus?

Lagrange property for Moufang loops

Does every finite Moufang loop have the strong Lagrange property?

Moufang loops with non-normal commutant

Is there a Moufang loop whose commutant is not normal?

Quasivariety of cores of Bol loops

Is the class of cores of Bol loops a quasivariety?

Parity of the number of quasigroups up to isomorphism

Let I(n) be the number of isomorphism classes of quasigroups of order n. Is I(n) odd for every n?

Classification of finite simple paramedial quasigroups

Classify the finite simple paramedial quasigroups.

See also

References

External links

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