Rocky Mountain Journal of Mathematics

Lagrange's theorem for Hom-Groups

Mohammad Hassanzadeh

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Hom-groups are nonassociative generalizations of groups where the unitality and associativity are twisted by a map. We show that a Hom-group $(G, \alpha )$ is a pointed idempotent quasigroup (pique). We use Cayley tables of quasigroups to introduce some examples of Hom-groups. Introducing the notions of Hom-subgroups and cosets we prove Lagrange's theorem for finite Hom-groups. This states that the order of any Hom-subgroup $H$ of a finite Hom-group $G$ divides the order of $G$. We linearize Hom-groups to obtain a class of nonassociative Hopf algebras called Hom-Hopf algebras. As an application of our results, we show that the dimension of a Hom-sub-Hopf algebra of the finite dimensional Hom-group Hopf algebra $\mathbb {K}G$ divides the order of $G$. The new tools introduced in this paper could potentially have applications in theories of quasigroups, nonassociative Hopf algebras, Hom-type objects, combinatorics, and cryptography.

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Rocky Mountain J. Math., Volume 49, Number 3 (2019), 773-787.

First available in Project Euclid: 23 July 2019

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Hassanzadeh, Mohammad. Lagrange's theorem for Hom-Groups. Rocky Mountain J. Math. 49 (2019), no. 3, 773--787. doi:10.1216/RMJ-2019-49-3-773.

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  • N. Aizawa and H. Sato, q-deformation of the Virasoro algebra with central extension, Phys. Lett. B, 256 (1991), 185–190.
  • D. Bryant, M. Buchanan, and I.M. Wanless, The spectrum for quasigroups with cyclic automorphisms and additional symmetries, Discrete Mathematics, 309 (2009), no. 4, 821–833.
  • R.H. Bruck, A survey of binary systems, 3rd ed., Springer (1971).
  • B. Baumeister and A. Stein, The finite Bruck loops, J. Algebra, 330 (2011), 206–220.
  • S. Caenepeel, and I. Goyvaerts, Monoidal Hom-Hopf algebras, Comm. Algebra 39 (2011), no. 6, 2216–2240.
  • M. Chaichian, P. Kulish, and J. Lukierski, q-deformed Jacobi identity, q-oscillators and q-deformed infinite-dimensional algebras, Phys. Lett. B, 237 (1990), 401–406.
  • O. Chein, M.K. Kinyon, A. Rajah, and P. Vojtěchovský, Loops and the Lagrange property, Results. Math. 43 (2003), 74–78.
  • O. Chein, H. Pflugfelder, and J.D.H. Smith, Quasigroups and loops: theory and applications, Heldermann, Berlin (1990).
  • T.L. Curtright and C.K. Zachos, Deforming maps for quantum algebras, Phys. Lett. B, 243 (1990), 237–244.
  • Y. Chen, Z. Wang, and L. Zhang, Integrals for monoidal Hom-Hopf algebras and their applications, J. Math. Phys. 54 (2013), 073515.
  • J. Dénes and A. Keedwell, Latin squares and their applications, Academic Press, (1974).
  • T. Foguel, M.K. Kinyon, and J.D. Phillips, On twisted subgroups and Bol loops of odd order, Rocky Mountain J. Math. 36 (2006), no. 1, 183–212.
  • G. Graziani, A. Makhlouf, C. Menini, and F. Panaite, BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras, SIGMA, 11 (2015), 086, 34 pages.
  • A.N. Grishkov and A.V. Zavarnitsine, Lagrange's theorem for Moufang loops, Math. Proc. Cambridge Philos. Soc. 139 (2005), 41–57.
  • M. Hassanzadeh, Hom-groups, Representations and homological algebra, arXiv 1801.07398. To appear in Colloq. Math.
  • M. Hassanzadeh, On antipodes of Hom-Hopf algebras (2018), arXiv 1803.01441.
  • J.T. Hartwig, D. Larsson, and S.D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivations, J. Algebra 295 (2006), no. 2, 314–361.
  • M. Hassanzadeh, I. Shapiro and S. Sütlü, Cyclic homology for Hom-associative algebras, J. Geom. and Phys., 98, December (2015), 40–56.
  • C. Laurent-Gengoux, A. Makhlouf, and J. Teles, Universal algebra of a Hom-Lie algebra and group-like elements, J. Pure Appl. Algebra, 222 (2018), no. 5, 1139–1163.
  • A. Makhlouf and S.D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2008), no. 2, 51–64.
  • A. Makhlouf and S. Silvestrov, Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras, pp. 189–206 in Generalized Lie theory in mathematics, physics and beyond, Springer, Berlin (2009)
  • A. Makhlouf and S. Silvestrov, Hom-algebras and Hom-coalgebras, J. Algebra Appl. 9 (2010), no. 4, 553–589.
  • F. Panaite, P. Schrader, and M.D. Staic, Hom-Tensor categories and the Hom-Yang-Baxter equation, Appl. Categ. Structures, 27 (2019), p 323–363.
  • B. Schneier, Applied cryptography, Wiley, New York (1996).
  • J.D.H. Smith, An introduction to quasigroups and their representations, chapman and Hall/CRC Press (2007).
  • T. Suksumran and K. Wiboonton, Lagrange's theorem for gyrogroups and the Cauchy property, Quasigroups Rel. Syst. 22 (2014), 283–294.
  • D. Yau, Hom-bialgebras and comodule Hom-algebras, Int. Electron. J. Algebra 8 (2010), 45–64.
  • D. Yau, Hom-quantum groups: I. Quasi-triangular Hom-bialgebras, J. Phys. 45 (2012) no. 6.
  • D. Yau, Enveloping algebras of Hom-Lie algebras, J. Gen. Lie Theory Appl., 2, (2008), 95–108.
  • X. Zhao, and X. Zhang, Lazy 2-cocycles over monoidal Hom-Hopf algebras, Colloq. Math. 142 (2016), no. 1, 61–81.