List of problems in loop theory and quasigroup theory

From HandWiki

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

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 [math]\displaystyle{ G }[/math], define [math]\displaystyle{ M(G,2) }[/math] on [math]\displaystyle{ G }[/math] x [math]\displaystyle{ C_2 }[/math] by [math]\displaystyle{ (g,0)(h,0)=(gh,0) }[/math], [math]\displaystyle{ (g,0)(h,1)=(hg,1) }[/math], [math]\displaystyle{ (g,1)(h,0)=(gh^{-1},1) }[/math], [math]\displaystyle{ (g,1)(h,1)=(h^{-1}g,0) }[/math]. Find a minimal presentation for the Moufang loop [math]\displaystyle{ M(G,2) }[/math] with respect to a presentation for [math]\displaystyle{ G }[/math].

  • Proposed: by Petr Vojtěchovský at Loops '03, Prague 2003
  • Comments: Chein showed in (Chein, 1974) that [math]\displaystyle{ M(G,2) }[/math] is a Moufang loop that is nonassociative if and only if [math]\displaystyle{ G }[/math] is nonabelian. Vojtěchovský (Vojtěchovský, 2003) found a minimal presentation for [math]\displaystyle{ M(G,2) }[/math] when [math]\displaystyle{ G }[/math] is a 2-generated group.

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?

  • Proposed: by Andrew Rajah at Loops '99, Prague 1999
  • Comments: The former has been solved by Rajah and Chee (2011) where they showed that for distinct odd primes p1 < ··· < pm < q < r1 < ··· < rn, all Moufang loops of order p12···pm2q3r12···rn2 are groups if and only if q is not congruent to 1 modulo pi for each i.

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

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

  • Proposed: by Andrew Rajah at Loops '03, Prague 2003

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. Nagy. The case n = pm holds by the Grishkov–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

References

  • Chein, Orin (1974), "Moufang Loops of Small Order I", Transactions of the American Mathematical Society 188: 31–51, doi:10.2307/1996765 .
  • Chein, Orin; Kinyon, Michael K.; Rajah, Andrew; Vojtěchovský, Petr (2003), "Loops and the Lagrange property", Results in Mathematics 43: 74–78, doi:10.1007/bf03322722 .
  • Chein, Orin; Rajah, Andrew (2000), "Possible orders of nonassociative Moufang loops", Commentationes Mathematicae Universitatis Carolinae 41 (2): 237–244 .
  • Conselo, E.; Conzales, S.; Markov, V.; Nechaev, A. (1998), "Recursive MDS-codes and recursively differentiable quasigroups", Diskretnaia Matematika 10 (2): 3–29 .
  • Daly, Dan; Vojtěchovský, Petr (2009), "Enumeration of nilpotent loops via cohomology", Journal of Algebra 322 (11): 4080–4098, doi:10.1016/j.jalgebra.2009.03.042 .
  • Drápal, Aleš (1992), "How far apart can the group multiplication tables be?", European Journal of Combinatorics 13 (5): 335–343, doi:10.1016/S0195-6698(05)80012-5 .
  • Drápal, Aleš (2000), "Non-isomorphic 2-groups coincide in at most three quarters of their multiplication tables", European Journal of Combinatorics 21 (3): 301–321, doi:10.1006/eujc.1999.0347 
  • Gagola III, Stephen (2012), "A Moufang loop's commutant", Mathematical Proceedings of the Cambridge Philosophical Society 152 (2): 193–206, doi:10.1017/S0305004111000181, Bibcode: 2012MPCPS.152..193G 
  • Grishkov, Alexander N.; Zavarnitsine, Andrei V. (2005), "Lagrange's theorem for Moufang loops", Mathematical Proceedings of the Cambridge Philosophical Society 139 (1): 41–57, doi:10.1017/S0305004105008388, Bibcode: 2005MPCPS.139...41G 
  • Grishkov, Alexander N.; Kinyon, Michael; Nagý, Gabor (2013), "Solvability of commutative automorphic loops", Proceedings of the American Mathematical Society 142 (9): 3029–3037, doi:10.1090/s0002-9939-2014-12053-3 
  • Kinyon, Michael K., A survey of Osborn loops, invited talk at Milehigh conference on quasigroups, loops and nonassociative systems, Denver, 2005, http://web.cs.du.edu/~petr/milehigh/2005/kinyon_talk.pdf 
  • Kinyon, Michael; Kunen, Kenneth; Phillips, J.D.; Vojtěchovský, Petr (2016), "The structure of automorphic loops", Transactions of the American Mathematical Society 368: 8901–8927, doi:10.1090/tran/6622 
  • Liebeck, M. W. (1987), "The classification of finite simple Moufang loops", Mathematical Proceedings of the Cambridge Philosophical Society 102 (1): 33–47, doi:10.1017/S0305004100067025, Bibcode: 1987MPCPS.102...33L 
  • Nagy, Gábor P. (2002), "The Campbell–Hausdorff series of local analytic Bruck loops", Abh. Math. Sem. Univ. Hamburg 72 (1): 79–87, doi:10.1007/BF02941666 .
  • Nagy, Gábor P.; Vojtěchovský, Petr (2007), "Moufang loops of order 64 and 81", Journal of Symbolic Computation (to appear) 42 (9): 871–883, doi:10.1016/j.jsc.2007.06.004 .
  • Nagy, Gábor P. (2008), "A class of simple proper Bol loops", Manuscripta Mathematica 127 (1): 81–88, doi:10.1007/s00229-008-0188-5 .
  • Nagy, Gábor P. (2009), "A class of finite simple Bol loops of exponent 2", Transactions of the American Mathematical Society 361 (10): 5331–5343, doi:10.1090/S0002-9947-09-04646-7 .
  • Niemenmaa, Markku (2009), "Finite loops with nilpotent inner mapping groups are centrally nilpotent", Bulletin of the Australian Mathematical Society 79 (1): 109–114, doi:10.1017/S0004972708001093 
  • Ormes, Nicholas; Vojtěchovský, Petr (2007), "Powers and alternative laws", Commentationes Mathematicae Universitatis Carolinae 48 (1): 25–40 .
  • Paige, L. (1956), "A class of simple Moufang loops", Proceedings of the American Mathematical Society 7 (3): 471–482, doi:10.2307/2032757 .
  • Rajah, Andrew; Chee, Wing Loon (2011), "Moufang loops of odd order p12p22···pn2q3", International Journal of Algebra 5 (20): 965–975 .
  • Rivin, Igor; Vardi, Ilan; Zimmerman, Paul (1994), "The n-queens problem", American Mathematical Monthly 101 (7): 629–639, doi:10.2307/2974691 .
  • Romanowska, Anna; Smith, Jonathan D. H. (1985), Modal Theory, Heldermann Verlag, Berlin .
  • Rozskowska-Lech, B. (1999), "A representation of symmetric idempotent and entropic groupoids", Demonstr. Math. 32: 248–262 .
  • Shcherbacov, V.A.; Pushkashu, D.I. (2010), "On the structure of finite paramedial quasigroups", Comment. Math. Univ. Carolin. 51: 357–370 .
  • Stones, D. S. (2010), "The parity of the number of quasigroups", Discrete Mathematics 310 (21): 3033–3039, doi:10.1016/j.disc.2010.06.027 .
  • Vojtěchovský, Petr (2003), "Generators for finite simple Moufang loops", Journal of Group Theory 6 (2): 169–174, doi:10.1515/jgth.2003.012 .
  • Vojtěchovský, Petr (2003), "The smallest Moufang loop revisited", Results in Mathematics 44: 189–193, doi:10.1007/bf03322924 .
  • Vojtěchovský, Petr (2006), "Toward the classification of Moufang loops of order 64", European Journal of Combinatorics 27 (3): 444–460, doi:10.1016/j.ejc.2004.10.003 .

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