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 | References | Similar Articles | Additional Information

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.

**1.**R. H. Bruck,*A Survey of Binary Systems*, Springer-Verlag, 1971. MR**20:76****2.**B. F. Bryant and H. Schneider, Principal loop-isotopes of quasigroups,*Canadian J. Math*18 (1966) 120 - 125. MR**32:5772****3.**C. C. Chang and H. J. Keisler,*Model Theory*, North-Holland, 1990. MR**91c:03026****4.**O. Chein,*Moufang loops of small order*, Memoirs Amer. Math. Soc. 13 (1978), no. 197. MR**57:6271****5.**O. Chein, H. O. Pflugfelder, and J. D. H. Smith,*Quasigroups and Loops: Theory and Applications*, Heldermann Verlag, 1990. MR**93g:20133****6.**T. Evans, Embedding Incomplete Latin Squares,*Amer. Math. Monthly*67 (1960) 958 - 961. MR**23:A68****7.**F. Fenyves, Extra Loops I,*Publicationes Mathematicae Debrecen*15 (1968) 235 - 238. MR**38:5976****8.**F. Fenyves, Extra Loops II,*Publicationes Mathematicae Debrecen*16 (1969) 187 - 192. MR**41:7017****9.**E. G. Goodaire and D. A. Robinson, Loops Which Are Cyclic Extensions of Their Nuclei,*Compositio Math.*45 (1982) 341 - 356. MR**83g:20079****10.**E. G. Goodaire and D. A. Robinson, A Class of Loops Which Are Isomorphic to All Loop Isotopes,*Canadian J. Math*34 (1982) 662 - 672. MR**83k:20079****11.**E. G. Goodaire and D. A. Robinson, Some Special Conjugacy Closed Loops,*Canadian Math Bull.*33 (1990) 73 - 78. MR**91a:20077****12.**J. Hart and K. Kunen, Single Axioms for Odd Exponent Groups,*J. Automated Reasoning*14 (1995) 383 - 412. MR**96h:68178****13.**K. Kunen, Moufang Quasigroups,*J. Algebra*183 (1996) 231-234. MR**97f:20096****14.**K. Kunen, Quasigroups, Loops, and Associative Laws,*J. Algebra*185 (1996) 194-204. MR**97g:20083****15.**W. W. McCune, OTTER 3.0 Reference Manual and Guide, Technical Report ANL-94/6, Argonne National Laboratory, 1994; available at URL:`http://www.mcs.anl.gov`

**16.**H. O. Pflugfelder,*Quasigroups and Loops: Introduction*, Heldermann Verlag, 1990. MR**93g:20132****17.**E. L. Wilson, A class of loops with the isotopy-isomorphy property,*Canadian J. Math*18 (1966) 589 - 592. MR**33:5779****18.**R. L. Wilson, Jr., Isotopy-isomorphy loops of prime order,*J. Algebra*31 (1974) 117 - 119. MR**49:10808****19.**R. L. Wilson, Jr., Quasidirect products of quasigroups,*Comm. Algebra*3 (1975) 835 - 850. MR**51:13112****20.**J. Zhang and H. Zhang, SEM: a system for enumerating models,*Proc. 14th Internat. Joint Conference on AI (IJCAI-95)*, Montréal, 1995, pp. 298 - 303; available at URL:`http://www.cs.uiowa.edu/~hzhang/`

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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