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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Structural interactions of conjugacy closed loops

Author: Ales Drápal
Journal: Trans. Amer. Math. Soc. 360 (2008), 671-689
MSC (2000): Primary 20N05; Secondary 08A05
Published electronically: September 4, 2007
MathSciNet review: 2346467
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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$.

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Ales Drápal
Affiliation: Department of Mathematics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Keywords: Conjugacy closed loop, multiplication group, nucleus
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.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.