Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T01:51:06.511Z Has data issue: false hasContentIssue false
  • English
  • Français

Searching for small simple automorphic loops

Part of: Permutation groups Other generalizations of groups

Published online by Cambridge University Press:  01 August 2011

Kenneth W. Johnson ,
Michael K. Kinyon ,
Gábor P. Nagy  and
Petr Vojtěchovský
Kenneth W. Johnson
Affiliation:
Penn State Abington, 1600 Woodland Rd, Abington PA 19001, USA (email: [email protected])
Michael K. Kinyon
Affiliation:
Department of Mathematics, University of Denver, 2360 S Gaylord St, Denver Colorado 80112, USA (email: [email protected])
Gábor P. Nagy
Affiliation:
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1 H-6720 Szeged, Hungary (email: [email protected])
Petr Vojtěchovský
Affiliation:
Department of Mathematics, University of Denver, 2360 S Gaylord St, Denver Colorado 80112, USA (email: [email protected])

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no non-associative simple commutative automorphic loop of order less than 212, and no non-associative simple automorphic loop of order less than 2500. We obtain numerous examples of non-associative simple right automorphic loops. We also prove that every automorphic loop has the antiautomorphic inverse property, and that a right automorphic loop is automorphic if and only if its conjugations are automorphisms.

MSC classification

Secondary: 20N05: Loops, quasigroups 20B15: Primitive groups 20B40: Computational methods
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 200 - 213
Copyright
Copyright © London Mathematical Society 2011

References

[1]Albert, A. A., ‘Quasigroups I’, Trans. Amer. Math. Soc. 54 (1943) 507–519.CrossRefGoogle Scholar
[2]Aschbacher, M., ‘On Bol loops of exponent 2’, J. Algebra 288 (2005) no. 1, 99–136.CrossRefGoogle Scholar
[3]Baer, R., ‘Nets and groups’, Trans. Amer. Math. Soc. 47 (1939) 110–141.CrossRefGoogle Scholar
[4]Baumeister, B. and Stein, A., ‘Self-invariant 1-factorizations of complete graphs and finite Bol loops of exponent 2’, Beiträge Algebra Geom. 51 (2010) no. 1, 117–135.Google Scholar
[5]Bruck, R. H., A survey of binary systems, Ergebnisse der Mathematik und ihrer Grenzgebiete 20 (Springer, Berlin, 1971) third printing, corrected.CrossRefGoogle Scholar
[6]Bruck, R. H. and Paige, L. J., ‘Loops whose inner mappings are automorphisms’, Ann. of Math. (2) 63 (1956) 308–323.CrossRefGoogle Scholar
[7]Cameron, P. J., ‘Almost all quasigroups have rank 2’, Discrete Math. 106/107 (1992) 111–115.CrossRefGoogle Scholar
[8]Conway, J. H., ‘The Golay codes and Mathieu groups’, Sphere packings, lattices and groups (eds Conway, J. H. and Sloane, N. J. A.; Springer, Berlin–New York, 1988) Chapter 11.CrossRefGoogle Scholar
[9]Drápal, A., Latin squares and groups, MS Thesis, Charles University, Prague, 1979.Google Scholar
[10]Drápal, A., ‘Multiplication groups of finite loops that fix at most two points’, J. Algebra 235 (2001) 154–175.CrossRefGoogle Scholar
[11]Drápal, A., ‘Multiplication groups of loops and projective semilinear transformations in dimension two’, J. Algebra 251 (2002) 256–278.CrossRefGoogle Scholar
[12]Drápal, A. and Kepka, T., ‘Alternating groups and latin squares’, European J. Combin. 10 (1989) 175–180.CrossRefGoogle Scholar
[13]Funk, M. and Nagy, P. T., ‘On collineation groups generated by Bol reflections’, J. Geom. 48 (1993) no. 1–2, 63–78.CrossRefGoogle Scholar
[14] The GAP group, GAP—groups, algorithms, and programming, version 4.4.12, 2008, available athttp://www.gap-system.org.Google Scholar
[15]Goodaire, E. G. and Robinson, D. A., ‘A class of loops which are isomorphic to all loop isotopes’, Canad. J. Math. 34 (1982) 662–672.CrossRefGoogle Scholar
[16]Goodaire, E. G. and Robinson, D. A., ‘Semi-direct products and Bol loop’, Demonstratio Math. 27 (1994) no. 3–4, 573–588.Google Scholar
[17]Guralnick, R. M. and Saxl, J., ‘Monodromy groups of polynomials’, Groups of Lie type and their geometries (Como, 1993), London Mathematical Society Lecture Note Series 207 (Cambridge University Press, Cambridge, 1995) 125–150.CrossRefGoogle Scholar
[18]Jedlička, P., Kinyon, M. K. and Vojtěchovský, P., ‘Constructions of commutative automorphic loops’, Comm. Algebra 38 (2010) no. 9, 3243–3267.CrossRefGoogle Scholar
[19]Jedlička, P., Kinyon, M. K. . and Vojtěchovský, P., ‘The structure of commutative automorphic loops’, Trans. Amer. Math. Soc. 363 (2011) 365–384.CrossRefGoogle Scholar
[20]Kiechle, H., Theory of K-loops, Lecture Notes in Mathematics 1778 (Springer, Berlin, 2002).CrossRefGoogle Scholar
[21]Kinyon, M. K., Kunen, K., Phillips, J. D. and Vojtěchovský, P., ‘The structure of automorphic loops’, Preprint.Google Scholar
[22]Kreuzer, A., ‘Inner mappings of Bruck loops’, Math. Proc. Cambridge Philos. Soc. 123 (1998) no. 1, 53–57.CrossRefGoogle Scholar
[23]Mazur, M., ‘Connected transversals to nilpotent groups’, J. Group Theory 2 (2007) 195–203.Google Scholar
[24]Nagy, G. P., ‘A class of finite simple Bol loops of exponent 2’, Trans. Amer. Math. Soc. 361 (2009) no. 10, 5331–5343.CrossRefGoogle Scholar
[25]Nagy, G. P., ‘On the multiplication groups of semifields’, European J. Combin. 31 (2010) no. 1, 18–24.CrossRefGoogle Scholar
[26]Nagy, G. P. and Vojtěchovský, P., ‘Loops: computing with quasigroups and loops in GAP, version 2.1.0’, available at http://www.math.du.edu/loops.Google Scholar
[27]Niemenmaa, M., ‘Finite loops with nilpotent inner mapping groups are centrally nilpotent’, Bull. Aust. Math. Soc. 79 (2009) 109–114.CrossRefGoogle Scholar
[28]Niemenmaa, M. and Kepka, T., ‘On multiplication groups of loops’, J. Algebra 135 (1990) no. 1, 112–122.CrossRefGoogle Scholar
[29]Niemenmaa, M. and Kepka, T., ‘On connected transversals to abelian subgroups’, Bull. Aust. Math. Soc. 49 (1994) no. 1, 121–128.CrossRefGoogle Scholar
[30]Roney-Dougal, C. M., ‘The primitive permutation groups of degree less than 2500’, J. Algebra 292 (2005) no. 1, 154–183.CrossRefGoogle Scholar
[31]Soicher, L. H., ‘GRAPE, Graph algorithms using permutation groups, version 4.3’, package for GAP, available at http://www.maths.qmul.ac.UK/∼leonard/grape/.Google Scholar
[32]Vesanen, A., ‘The group PSL(2,q) is not the multiplication group of a loop’, Comm. Algebra 22 (1994) 1177–1195.CrossRefGoogle Scholar
[33]Vesanen, A., ‘Finite classical groups and multiplication groups of loops’, Math. Proc. Cambridge Philos. Soc. 117 (1995) 425–429.CrossRefGoogle Scholar
[34]Vesanen, A., ‘Solvable groups and loops’, J. Algebra 180 (1996) no. 3, 862–876.CrossRefGoogle Scholar