Structural interactions of conjugacy closed loops
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Abstract:
We study conjugacy closed loops by means of their multiplication groups. Let $Q$ be a conjugacy closed loop, $N$ its nucleus, $A$ the associator subloop, and $\mathcal L$ and $\mathcal R$ the left and right multiplication groups, respectively. Put $M = \{a\in Q;$ $L_a \in \mathcal R\}$. We prove that the cosets of $A$ agree with orbits of $[\mathcal L, \mathcal R]$, that $Q/M \cong (\operatorname {Inn} Q)/\mathcal L_1$ and that one can define an abelian group on $Q/N \times \operatorname {Mlt}_1$. We also explain why the study of finite conjugacy closed loops can be restricted to the case of $N/A$ nilpotent. Group $[\mathcal {L},\mathcal {R}]$ is shown to be a subgroup of a power of $A$ (which is abelian), and we prove that $Q/N$ can be embedded into $\operatorname {Aut}([\mathcal {L}, \mathcal {R}])$. Finally, we describe all conjugacy closed loops of order $pq$.References
- A. S. Basarab, A class of $\textrm {LK}$-loops, Mat. Issled. 120, Bin. i $n$-arnye Kvazigruppy (1991), 3–7, 118 (Russian). MR 1121425
- V. D. Belousov, Osnovy teorii kvazigrupp i lup, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0218483
- Richard Hubert Bruck, A survey of binary systems, Reihe: Gruppentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0093552, DOI 10.1007/978-3-662-35338-7
- Aleš Drápal, Conjugacy closed loops and their multiplication groups, J. Algebra 272 (2004), no. 2, 838–850. MR 2028083, DOI 10.1016/j.jalgebra.2003.06.011
- Aleš Drápal, Multiplication groups of finite loops that fix at most two points, J. Algebra 235 (2001), no. 1, 154–175. MR 1807660, DOI 10.1006/jabr.2000.8472
- Aleš Drápal, Orbits of inner mapping groups, Monatsh. Math. 134 (2002), no. 3, 191–206. MR 1883500, DOI 10.1007/s605-002-8256-2
- Edgar G. Goodaire and D. A. Robinson, A class of loops which are isomorphic to all loop isotopes, Canadian J. Math. 34 (1982), no. 3, 662–672. MR 663308, DOI 10.4153/CJM-1982-043-2
- Edgar G. Goodaire and D. A. Robinson, Some special conjugacy closed loops, Canad. Math. Bull. 33 (1990), no. 1, 73–78. MR 1036860, DOI 10.4153/CMB-1990-013-9
- Tomáš Kepka and Markku Niemenmaa, On loops with cyclic inner mapping groups, Arch. Math. (Basel) 60 (1993), no. 3, 233–236. MR 1201636, DOI 10.1007/BF01198806
- Kenneth Kunen, The structure of conjugacy closed loops, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2889–2911. MR 1615991, DOI 10.1090/S0002-9947-00-02350-3
- Michael K. Kinyon, Kenneth Kunen, and J. D. Phillips, Diassociativity in conjugacy closed loops, Comm. Algebra 32 (2004), no. 2, 767–786. MR 2101839, DOI 10.1081/AGB-120027928
- Markku Niemenmaa and Tomáš Kepka, On multiplication groups of loops, J. Algebra 135 (1990), no. 1, 112–122. MR 1076080, DOI 10.1016/0021-8693(90)90152-E
- L. V. Sabinin, The geometry of loops, Mat. Zametki 12 (1972), 605–616 (Russian). MR 340461
- Lev V. Sabinin, Smooth quasigroups and loops, Mathematics and its Applications, vol. 492, Kluwer Academic Publishers, Dordrecht, 1999. MR 1727714, DOI 10.1007/978-94-011-4491-9
- V. D. Belousov (ed.), Voprosy teorii kvazigrupp i lup, Redakcionno-Izdat. Otdel Akad. Nauk Moldav. SSR, Kishinev, 1970, 1971 (Russian). MR 0274626
Additional Information
- Aleš Drápal
- Affiliation: Department of Mathematics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
- Email: drapal@karlin.mff.cuni.cz
- Received by editor(s): June 3, 2003
- Received by editor(s) in revised form: August 29, 2005
- Published electronically: September 4, 2007
- Additional Notes: The author was supported by institutional grant MSM 0021620839.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 671-689
- MSC (2000): Primary 20N05; Secondary 08A05
- DOI: https://doi.org/10.1090/S0002-9947-07-04131-1
- MathSciNet review: 2346467