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? Proposed: by Michael Kinyon, based on (Chein and Rajah, 2000) Comments: The assumption that L/M has order bigger than 3 is important, as there is a (commutative) Moufang loop L of order 81 with normal commutative subgroup of order 27.
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. Proposed: by Alexander Grishkov at Loops '03, Prague 2003 Comment: Grishkov and Shestakov, 2011 proved that for n<7, the n-generated free commutative Moufang loop of exponent 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. Proposed: by Alexander Grishkov at Loops '11, Třešť 2011
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 . Proposed: by Petr Vojtěchovský at Loops '03, Prague 2003 Comments: Chein showed in (Chein, 1974) that is a Moufang loop that is nonassociative if and only if is nonabelian. Vojtěchovský (Vojtěchovský, 2003) found a minimal presentation for when is a 2-generated group.
Presentations for finite simple Moufang loops Find presentations for all nonassociative finite simple Moufang loops in the variety of Moufang loops. Proposed: by Petr Vojtěchovský at Loops '03, Prague 2003 Comments: It is shown in (Vojtěchovský, 2003) that every nonassociative finite simple Moufang loop is generated by 3 elements, with explicit formulas for the generators.
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). Proposed: by Alexander Grishkov at Loops '11, Třešť 2011 Comments: In the case when n is a prime different from 3 the conjecture was proved by Grishkov. If p = 3 and M is commutative, it was proved by Bruck. The general case for p = 3 was proved by G. Nagyhttps://www.acta.hu/download.phtml?id=2632. The case n = pm holds by the Grishkov–Sabinina–Zelmanov Theorem.
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. Proposed: by Alexander Grishkov at Loops '11, Třešť 2011
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. Proposed: by Alexander Grishkov at Loops '03, Prague 2003
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. Proposed: at Milehigh conference on quasigroups, loops, and nonassociative systems, Denver 2005
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? Proposed: by Aleš Drápal at Loops '99, Prague 1999 Comments: Drápal proved in (Drápal, 1992) that groups are quadratic with , and in (Drápal, 2000) that 2-groups are quadratic with .
Campbell–Hausdorff series for analytic Bol loops Determine the Campbell–Hausdorff series for analytic Bol loops. Proposed: by M. A. Akivis and V. V. Goldberg at Loops '99, Prague 1999 Comments: The problem has been partially solved for local analytic Bruck loops in (Nagy, 2002).
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−1x−1. Is there a finite, universally flexible loop that is not middle Bol? Proposed: by Michael Kinyon at Loops '03, Prague 2003
Finite simple Bol loop with nontrivial conjugacy classes Is there a finite simple nonassociative Bol loop with nontrivial conjugacy classes? Proposed: by Kenneth W. Johnson and Jonathan D. H. Smith at the 2nd Mile High Conference on Nonassociative Mathematics, Denver 2009
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? Proposed: at Loops '03 and '07, Prague 2003 and 2007 Comments: The answer to the first question is affirmative if Q is finite (Niemenmaa 2009). The problem is open in the general case.
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? Proposed: at Loops '07, Prague 2007 Comments: When the inner mapping group Inn(Q) is finite and abelian, then Q is nilpotent (Niemenaa and Kepka). The first question is therefore open only in the infinite case. Call loop Q of Csörgõ type if it is nilpotent of class at least 3, and Inn(Q) is abelian. No loop of Csörgõ type of nilpotency class higher than 3 is known. Loops of Csörgõ type exist (Csörgõ, 2004), Buchsteiner loops of Csörgõ type exist (Csörgõ, Drápal and Kinyon, 2007), and Moufang loops of Csörgõ type exist (Nagy and Vojtěchovský, 2007). On the other hand, there are no groups of Csörgõ type (folklore), there are no commutative Moufang loops of Csörgõ type (Bruck), and there are no Moufang p-loops of Csörgõ type for p > 3 (Nagy and Vojtěchovský, 2007).
Number of nilpotent loops up to isomorphism Determine the number of nilpotent loops of order 24 up to isomorphism. Proposed: by Petr Vojtěchovský at the 2nd Mile High Conference on Nonassociative Mathematics, Denver 2009 Comment: The counts are known for n < 24, see (Daly and Vojtěchovský, 2010). The number of nilpotent loops of order 24 with center of size at least 3 is 2,318,640,494,742,419,998,834, see .
A finite nilpotent loop without a finite basis for its laws Construct a finite nilpotent loop with no finite basis for its laws. Proposed: by M. R. Vaughan-Lee in the Kourovka Notebook of Unsolved Problems in Group Theory Comment: There is a finite loop with no finite basis for its laws (Vaughan-Lee, 1979) but it is not nilpotent.
Open problems (quasigroups)
Existence of infinite simple paramedial quasigroups Are there infinite simple paramedial quasigroups? Proposed: by Jaroslav Ježek and Tomáš Kepka at Loops '03, Prague 2003
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?
Proposed: by Tomáš Kepka and Petr Němec at Loops '03, Prague 2003 Comments: Every quasigroup is isotopic to a loop, hence the variety of loops is isotopically universal.
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?
Proposed: by Parascovia Syrbu at Loops '03, Prague 2003 Comments: see (Conselo et al., 1998)
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?
Proposed: by Gábor Nagy at the 2nd Mile High Conference on Nonassociative Mathematics, Denver 2009
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?
Proposed: at the Milehigh conference on quasigroups, loops, and nonassociative systems, Denver 2005
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?
Proposed: by Warren D. Smith Comments: There are infinite alternative loops without 2-sided inverses, cf. (Ormes and Vojtěchovský, 2007). It is known that every finite alternative loop of order at most 185 has 2-sided inverses, cf. https://www.academia.edu/102902796/Inclusions_among_diassociativity_related_loop_properties.
Finite simple nonassociative automorphic loop Find a nonassociative finite simple automorphic loop, if such a loop exists.
Proposed: by Michael Kinyon at Loops '03, Prague 2003 Comments: It is known that such a loop cannot be commutative (Grishkov, Kinyon and Nagý, 2013) nor have odd order (Kinyon, Kunen, Phillips and Vojtěchovský, 2013).
Universality of Osborn loops A loop is Osborn if it satisfies the identity . 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?
Proposed: by Michael Kinyon at Milehigh conference on quasigroups, loops, and nonassociative systems, Denver 2005 Comments: Moufang and conjugacy closed loops are Osborn. See (Kinyon, 2005) for more.
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?
Proposed: at Milehigh conference on quasigroups, loops, and nonassociative systems, Denver 2005 Solved by: Piroska Csörgõ, Aleš Drápal, and Michael Kinyon Solution: The quotient of a Buchsteiner loop by its nucleus is an abelian group of exponent 4. In particular, no nonassociative Buchsteiner loop can be simple. There exists a Buchsteiner loop of order 128 which is not conjugacy closed.
Classification of Moufang loops of order 64 Classify nonassociative Moufang loops of order 64.
Proposed: at Milehigh conference on quasigroups, loops, and nonassociative systems, Denver 2005 Solved by: Gábor P. Nagy and Petr Vojtěchovský Solution: There are 4262 nonassociative Moufang loops of order 64. They were found by the method of group modifications in (Vojtěchovský, 2006), and it was shown in (Nagy and Vojtěchovský, 2007) that the list is complete. The latter paper uses a linear-algebraic approach to Moufang loop extensions.
Conjugacy closed loop with nonisomorphic one-sided multiplication groups