Bi-Gyrogroup: The Group-Like Structure Induced by Bi-Decomposition of Groups

Document Type : Special Issue: AIMC 51


Department of Mathematics, North Dakota State University, Fargo, ND 58105, USA


‎The decomposition $\Gamma=BH$ of a group $\Gamma$ into a subset B ‎and a subgroup $H$ of $\Gamma$ induces‎, ‎under general conditions‎, ‎a ‎group-like structure for B‎, ‎known as a gyrogroup‎. ‎The famous‎ concrete realization of a gyrogroup‎, ‎which motivated the emergence ‎of gyrogroups into the mainstream‎, ‎is the space of all ‎relativistically admissible velocities along with a binary ‎\mbox{operation} given by the Einstein velocity addition law of ‎special relativity theory‎. ‎The latter leads to the Lorentz ‎transformation group $\so{1,n}$‎, ‎$n\in\N$‎, ‎in pseudo-Euclidean ‎spaces of signature $(1‎, ‎n)$‎. ‎The study in this article is motivated ‎by generalized Lorentz groups $\so{m‎, ‎n}$‎, ‎$m‎, ‎n\in\N$‎, ‎in ‎pseudo-Euclidean spaces of signature $(m‎, ‎n)$‎. ‎Accordingly‎, ‎this ‎article explores the bi-decomposition $\Gamma = H_LBH_R$ of a group ‎$\Gamma$ into a subset $B$ and subgroups $H_L$ and $H_R$ of ‎$\Gamma$‎, ‎along with the novel bi-gyrogroup structure of $B$ induced ‎by the bi-decomposition of $\Gamma$‎. ‎As an example‎, ‎we show by ‎methods of Clifford \mbox{algebras} that the quotient group of the ‎spin group $\spin{m‎, ‎n}$ possesses the bi-decomposition structure‎.


1. M. Aschbacher, Near subgroups of finite groups, J. Group Theory 1 (1998) 113–129.

2. M. Aschbacher, On Bol loops of exponent 2,
J. Algebra 288 (2005) 99–136.

3. M. Aschbacher, M. K. Kinyon, J. D. Phillips, Finite Bruck loops,
Trans. Amer. Math. Soc. 358(7) (2005) 3061–3075.

‎F. Chatelin‎, Qualitative Computing‎: ‎A Computational Journey into‎ Non-linearity, World Scientific Publishing‎, Hackensack‎, ‎NJ‎, ‎2012‎.
5. D. S. Dummit, R. M. Foote, Abstract Algebra, John Wiley & Sons, Hoboken, NJ, 3 edition, 2004.

6. T. Feder, Strong near subgroups and left gyrogroups,
J. Algebra 259 (2003) 177–190.

7. M. Ferreira, Factorizations of Möbius gyrogroups,
Adv. Appl. Clifford Algebras 19 (2009) 303–323.

‎M‎. ‎Ferreira‎, ‎Gyrogroups in Projective Hyperbolic Clifford Analysis‎, ‎ in‎: ‎Hypercomplex Analysis and Applications‎, ‎Trends in Mathematics‎,
I‎. ‎Sabadini‎, ‎F‎. ‎Sommen F‎. ‎(Eds.)‎, ‎Springer‎, ‎Basel‎, ‎2011‎.

‎M. Ferreira‎,
Harmonic analysis on the Einstein gyrogroup‎, J‎. ‎Geom‎. ‎Symmetry Phys. 35 (2014) 21-60‎.

10. M. Ferreira, Harmonic analysis on the Möbius gyrogroup,
J. Fourier Anal. Appl. 21(2) (2015) 281–317.

11. M. Ferreira, G. Ren, Möbius gyrogroups: A Clifford algebra approach,
J. Algebra 328 (2011) 230–253.

‎M. Ferreira‎, ‎F. Sommen‎,
Complex boosts‎: ‎ a Hermitian Clifford algebra approach‎, Adv‎. ‎Appl‎. ‎Clifford Algebras 23(2) (2013) 339-362‎.

13. T. Foguel, M. K. Kinyon, J. D. Phillips, On twisted subgroups and Bol loops of odd order,
Rocky Mountain J. Math. 36 (2006) 183–212.

14. T. Foguel, A. A. Ungar, Involutory decomposition of groups into twisted subgroups and subgroups,
J. Group Theory 3 (2000) 27–46.

‎J. E‎. ‎Gilbert‎, ‎M.~A‎. ‎Murray‎,
Clifford Algebras and Dirac Operators in Harmonic‎ Analysis‎, Cambridge University Press‎, ‎Cambridge‎, ‎1991‎.

16. L. C. Grove,
Classical Groups and Geometric Algebra, Volume 39 of Graduate Studies in Mathematics, AMS, Providence, RI, 2001.

‎T. G‎. ‎Jaiyéolá‎, ‎A. R. T‎. ‎Sòlárìn‎, ‎J. O‎. ‎Adéníran‎,
Some Bol-Moufang characterization of the Thomas precession of a‎ gyrogroup‎, Algebras Groups Geom. 31(3) (2014) 341-362‎.

18. R. Lal, A. Yadav, Topological right gyrogroups and gyrotransversals,
Comm. Algebra 41 (2013) 3559–3575.

19. H. B. Lawson, M.-L. Michelsohn,
Spin Geometry, Princeton University Press, Princeton, NJ, 1989.

20. J. Lawson, Clifford algebras, Möbius transformations, Vahlen matrices, and B-loops.
Comment. Math. Univ. Carolin. 51(2) (2010) 319–331.
21. P. Lounesto, Clifford Algebras and Spinors, London Mathematical Society Lecture Note Series. 286, Cambridge University Press, Cambridge, 2 edition, 2001.

22. ‎N. Sonmez‎, ‎A. A‎. ‎Ungar‎,
The Einstein relativistic velocity model of hyperbolic geometry and‎ its plane separation axiom‎, Adv‎. ‎Appl‎. ‎Clifford Algebras  23 (2013) 209-236‎.

23. T. Suksumran, K. Wiboonton, Lagrange’s theorem for gyrogroups and the Cauchy property,
Quasigroups Related Systems 22(2) (2014) 283–294.

24. T. Suksumran, K. Wiboonton, Einstein gyrogroup as a B-loop,
Rep. Math. Phys. 76 (2015) 63–74.

25. T. Suksumran, K. Wiboonton, Isomorphism theorems for gyrogroups and L-subgyrogroups,
J. Geom. Symmetry Phys. 37 (2015) 67–83.

26. A. A. Ungar, Thomas rotation and parametrization of the Lorentz transformation group,
Found. Phys. Lett. 1 (1988) 57–89.

27. A. A. Ungar, Thomas precession and its associated grouplike structure,
Amer. J. Phys. 59(9) (1991) 824–834.

28. A. A. Ungar,
Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces, Volume 117 of Fundamental Theories of Physics, Kluwer Academic, Dordrecht, 2001.

29. A. A. Ungar,
Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity, World Scientific, Hackensack, NJ, 2008.

30. A. A. Ungar, From Möbius to gyrogroups,
Amer. Math. Monthly 115(2) (2008) 138–144.

31. A. A. Ungar,
A Gyrovector Space Approach to Hyperbolic Geometry, Synthesis Lectures on Mathematics and Statistics #4. Morgan & Claypool, San Rafael, CA, 2009.

32. A. A. Ungar,
Barycentric Calculus in Euclidean and Hyperbolic Geometry: A Comparative Introduction, World Scientific, Hackensack, NJ, 2010.

33. A. A. Ungar,
Analytic Hyperbolic Geometry in n Dimensions: An Introduction, CRC Press, Boca Raton, FL, 2015.

34. A. A. Ungar, Parametric realization of the Lorentz transformation group in pseudo-Euclidean spaces,
J. Geom. Symmetry Phys. 38 (2015) 39–108.