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Transactions of the American Mathematical Society

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

 
 

 

The structure of conjugacy closed loops


Author: Kenneth Kunen
Journal: Trans. Amer. Math. Soc. 352 (2000), 2889-2911
MSC (2000): Primary 20N05; Secondary 03C05, 08A05
DOI: https://doi.org/10.1090/S0002-9947-00-02350-3
Published electronically: February 16, 2000
MathSciNet review: 1615991
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Abstract: We study structure theorems for the conjugacy closed (CC-) loops, a specific variety of G-loops (loops isomorphic to all their loop isotopes). These theorems give a description all such loops of small order. For example, if $p$ and $q$ are primes, $p < q$, and $q-1$ is not divisible by $p$, then the only CC-loop of order $pq$ is the cyclic group of order $pq$. For any prime $q > 2$, there is exactly one non-group CC-loop in order $2q$, and there are exactly three in order $q^2$. We also derive a number of equations valid in all CC-loops. By contrast, every equation valid in all G-loops is valid in all loops.


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Additional Information

Kenneth Kunen
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: kunen@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02350-3
Keywords: Conjugacy closed loop, G-loop, isotopy
Received by editor(s): September 27, 1996
Received by editor(s) in revised form: March 13, 1998
Published electronically: February 16, 2000
Additional Notes: Author supported by NSF Grants CCR-9503445 and DMS-9704520. The author is grateful to the referee for many helpful comments on the original draft of this paper.
Article copyright: © Copyright 2000 American Mathematical Society

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