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 and are primes, , and is not divisible by , then the only CC-loop of order is the cyclic group of order . For any prime , there is exactly one non-group CC-loop in order , and there are exactly three in order . 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