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 Construct a conjugacy closed loop whose left multiplication group is not isomorphic to its right multiplication group. •
Proposed: by Aleš Drápal at Loops '03, Prague 2003 •
Solved by: Aleš Drápal •
Solution: There is such a loop of order 9. In can be obtained in the LOOPS package by the command
Existence of a finite simple Bol loop Is there a finite simple
Bol loop that is not Moufang? •
Proposed at: Loops '99, Prague 1999 •
Solved by: Gábor P. Nagy, 2007. •
Solution: A simple Bol loop that is not Moufang will be called
proper. • : There are several families of proper simple Bol loops. A smallest proper simple Bol loop is of order 24 (Nagy 2008). • : There is also a proper simple Bol loop of exponent 2 (Nagy 2009), and a proper simple Bol loop of odd order (Nagy 2008). •
Comments: The above constructions solved two additional open problems: • Is there a finite simple Bruck loop that is not Moufang? Yes, since any proper simple Bol loop of exponent 2 is Bruck. • Is every Bol loop of odd order solvable? No, as witnessed by any proper simple Bol loop of odd order.
Left Bol loop with trivial right nucleus Is there a finite non-Moufang left
Bol loop with trivial right nucleus? •
Proposed: at Milehigh conference on quasigroups, loops, and nonassociative systems, Denver 2005 •
Solved by: Gábor P. Nagy, 2007 •
Solution: There is a finite simple left Bol loop of exponent 2 of order 96 with trivial right nucleus. Also, using an
exact factorization of the
Mathieu group M24, it is possible to construct a non-Moufang simple Bol loop which is a
G-loop.
Lagrange property for Moufang loops Does every finite Moufang loop have the strong Lagrange property? •
Proposed: by Orin Chein at Loops '99, Prague 1999 •
Solved by: Alexander Grishkov and Andrei Zavarnitsine, 2003 •
Solution: Every finite Moufang loop has the strong Lagrange property (SLP). Here is an outline of the proof: • According to (Chein et al. 2003), it suffices to show SLP for nonassociative finite simple Moufang loops (NFSML). • It thus suffices to show that the order of a maximal subloop of an NFSML L divides the order of L. • A countable class of NFSMLs M(q) was discovered in (Paige 1956), and no other NSFMLs exist by (Liebeck 1987). • Grishkov and Zavarnitsine matched maximal subloops of loops M(q) with certain subgroups of groups with
triality in (Grishkov and Zavarnitsine, 2003).
Moufang loops with non-normal commutant Is there a Moufang loop whose commutant is not normal? •
Proposed: by Andrew Rajah at Loops '03, Prague 2003 •
Solved by: Alexander Grishkov and Andrei Zavarnitsine, 2017 •
Solution: Yes, there is a Moufang loop of order 38 with non-normal commutant. Gagola had previously claimed the opposite, but later found a hole in his proof.
Quasivariety of cores of Bol loops Is the class of cores of Bol loops a quasivariety? •
Proposed: by Jonathan D. H. Smith and Alena Vanžurová at Loops '03, Prague 2003 •
Solved by: Alena Vanžurová, 2004. •
Solution: No, the class of cores of Bol loops is not closed under subalgebras. Furthermore, the class of cores of groups is not closed under subalgebras. Here is an outline of the proof: • Cores of abelian groups are
medial, by (Romanowska and Smith, 1985), (Rozskowska-Lech, 1999). • The smallest nonabelian group S_3 has core containing a sub
magma G of order 4 that is not medial. • If G is a core of a Bol loop, it is a core of a Bol loop of order 4, hence a core of an abelian group, a contradiction.
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? •
Proposed: by Douglas S. Stones at 2nd Mile High Conference on Nonassociative Mathematics, Denver 2009 •
Solved by: Douglas S. Stones, 2010. •
Solution: I(12) is even. In fact, I(n) is odd for all
n ≤ 17 except 12. (Stones 2010)
Classification of finite simple paramedial quasigroups Classify the finite simple paramedial quasigroups. •
Proposed: by Jaroslav Ježek and Tomáš Kepka at Loops '03, Prague 2003. •
Solved by: Victor Shcherbacov and Dumitru Pushkashu (2010). •
Solution: Any finite simple paramedial quasigroup is isotopic to
elementary abelian p-group. Such quasigroup can be either a medial unipotent quasigroup, or a medial commutative distributive quasigroup, or special kind isotope of (φ+ψ)-simple medial distributive quasigroup. == See also ==