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Abstract
We introduce a notion of natural orderings of elements of connected finite quandles. Let Q be such a quandle of order n. Any automophism on Q is a natural ordering when the elements are already ordered naturally. Suppose that the permutation *q is a cycle of length n − 1. Then, the operation tables for such orderings coincide, which leads to the classification of automorphisms on Q. Moreover, every row and column of the operation table contains all the elements of such a quandle, which is due to K. Oshiro. We also consider the general case of finite connected quandles.
ACKNOWLEDGMENT
The author would like to thank Chikara Nakayama and Kanako Oshiro for helpful comments. He is grateful to the referee for many pieces of advice to make this article readable.
The author is partially supported by Grant-in-Aid for Scientific Research (No. 22540101), Ministry of Education, Science, Sports and Technology, Japan.
Notes
Communicated by S. Hermiller.