Rocky Mountain Journal of Mathematics

On Twisted Subgroups and Bol Loops of Odd Order

Tuval Foguel, Michael K. Kinyon, and J.D. Phillips

Full-text: Open access

Article information

Source
Rocky Mountain J. Math., Volume 36, Number 1 (2006), 183-212.

Dates
First available in Project Euclid: 5 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1181069494

Digital Object Identifier
doi:10.1216/rmjm/1181069494

Mathematical Reviews number (MathSciNet)
MR2228190

Zentralblatt MATH identifier
1136.20053

Subjects
Primary: 20N05: Loops, quasigroups [See also 05Bxx]

Keywords
Bol loop twisted subgroup

Citation

Foguel, Tuval; Kinyon, Michael K.; Phillips, J.D. On Twisted Subgroups and Bol Loops of Odd Order. Rocky Mountain J. Math. 36 (2006), no. 1, 183--212. doi:10.1216/rmjm/1181069494. https://projecteuclid.org/euclid.rmjm/1181069494


Export citation

References

  • M. Aschbacher, Near subgroups of finite groups, J. Group Theory 1 (1998), 113-129.
  • --------, Bol loops of exponent $2$, J. Algebra, to appear.
  • R. Baer, Nets and groups, Trans. Amer. Math. Soc. 47 (1939), 110-141.
  • V.D. Belousov, The core of a Bol loop, in Collection in General Algebra (Sem), Akad. Nauk. Moldav. SSR, Kishinev, (1965), 53-66 (in Russian).
  • --------, Foundations of the theory of quasigroups and loops, Izdat. Nauka, Moscow, 1967 (in Russian).
  • L. Bénéteau, Commutative Moufang loops and related groupoids, Chapter IV in 9, 115-142.
  • R.H. Bruck, A survey of binary systems, Springer Verlag, Berlin, 1971.
  • O. Chein, M.K. Kinyon, A. Rajah, and P. Vojtěchovský, Loops and the Lagrange property, Results in Math. 43 (2003), 74-78.
  • O. Chein, H.O. Pflugfelder, and J.D.H. Smith, eds., Quasigroups and loops: Theory and applications, Sigma Ser. Pure Math., vol. 9, Heldermann, Berlin, 1990.
  • S. Doro, Simple Moufang loops, Math. Proc. Cambridge Philos. Soc. 83 (1978), 377-392.
  • T. Feder, Strong near subgroups and left gyrogroups, J. Algebra 259 (2003), 177-190.
  • T. Feder and M. Vardi, The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory, SIAM J. Comput. 28 (1998) 57-104.
  • W. Feit and J. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775-1029.
  • B. Fischer, Distributive Quasigruppen endlicher Ordnung, Math. Zeit. 83, (1964) 267-303.
  • T. Foguel, Groups with polar decompositions, Results in Math. 42 (2002), 69-73.
  • T. Foguel and A.A. Ungar, Involutory decomposition of groups into twisted subgroups and subgroups, J. Group Theory 3 (2000), 27-46.
  • M. Funk and P.T. Nagy, On collineation groups generated by Bol reflections, J. Geometry 48 (1993) 63-78.
  • G. Glauberman, On loops of odd order I, J. Algebra 1 (1964), 374-396.
  • --------, On loops of odd order II, J. Algebra 8 (1968), 393-414.
  • --------, Central elements in core-free groups, J. Algebra 4 (1966), 403-420.
  • D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg. 23 (1982), 37-66.
  • H. Kiechle, Theory of K-loops, Lecture Notes in Math., vol. 1778, Springer-Verlag, Berlin, 2002.
  • M. Kikkawa, On some quasigroups of algebraic models of symmetric spaces II, Mem. Fac. Lit. Sci., Shimane Univ., Nat. Sci. 7 (1974), 29-35.
  • M.K. Kinyon and O. Jones, Loops and semidirect products, Comm. Algebra 28 (2000), 4137-4164.
  • A. Kreuzer, Beispiele endlicher und unendlicher K-loops, Results in Math. 23 (1993), 355-362.
  • O. Loos, Symmetric spaces I, J. Benjamin, New York, 1969.
  • G.P. Nagy, Group invariants of certain Burn loop classes, Bull. Belg. Math. Soc. 5 (1998), 403-415.
  • --------, Bol-tükrözések alkalmazásai, Ph.D. Dissertation, Bolyai Institute, Szeged, 1999 (in Hungarian).
  • P.T. Nagy and K. Strambach, Loops, their cores and symmetric spaces, Israel J. Math. 105 (1998), 285-322.
  • N. Nobusawa, Orthogonal groups and symmetric sets, Osaka J. Math. 20 (1983), 5-8.
  • H.O. Pflugfelder, Quasigroups and loops: An introduction, Sigma Ser. Pure Math., vol. 8, Heldermann, Berlin, 1990.
  • J.D. Phillips, Quotients of groups, Tamkang J. Math. 28 (1997), 271-275.
  • R.S. Pierce, Symmetric groupoids, Osaka J. Math. 15 (1978), 51-76.
  • --------, Symmetric groupoids II, Osaka J. Math. 16 (1979), 317-348.
  • Problems in loop theory and quasigroup theory, available at http://adela.karlin.mff.cuni.cz/$^\sim$loops03/plq/main.html
  • D.A. Robinson, Bol loops, Trans. Amer. Math. Soc. 123 (1966), 341-354.
  • D.A. Robinson and K. Robinson, A class of Bol loops whose nuclei are not normal, Arch. Math. (Basel) 61 (1993), 596-600.
  • N. Umaya, On symmetric structure of a group, Proc. Japan Acad. 52 (1976), 174-176.
  • A.A. Ungar, Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics, Found. Phys. 27 (1997), 881-951.