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 formula_1, define formula_2 on formula_1 x formula_4 by
formula_5, formula_6, formula_7, formula_8. Find a minimal presentation for the Moufang loop formula_2 with respect to a presentation for formula_1.
Moufang loops of order "p"2"q"3 and "pq"4.
Let "p" and "q" be distinct odd primes. If "q" is not congruent to 1 modulo "p", are all Moufang loops of order "p"2"q"3 groups? What about "pq"4?
(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 MF"n" be the free Moufang loop with "n" generators.
Conjecture: MF3 is torsion free but MF"n" 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 formula_15, formula_16 be two quasigroups defined on the same underlying set formula_17. The distance formula_18 is the number of pairs formula_19 in formula_20 such that formula_21. Call a class of finite quasigroups "quadratic" if there is a positive real number formula_22 such that any two quasigroups formula_15, formula_16 of order formula_25 from the class satisfying formula_26 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 = "y"−1"x"−1. 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.
A finite nilpotent loop without a finite basis for its laws.
Construct a finite nilpotent loop with no finite basis for its laws.
Open problems (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" = "K""n","n" be the complete bipartite graph with distinct weights on its "n"2 edges. Let "M"1 be the cheapest matching in "G", "M"2 the cheapest matching in "G" with "M"1 removed, and so on. Each matching "M""i" determines a permutation "p""i" of 1, ..., "n". Let "L" be obtained from "G" by placing the permutation "p""i" 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") = ("x""λ"\"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.
References.
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