Contract grant sponsor: Australian Research Council; contract grant numbers: DP1093320 and FT110100065; contract grant numbers: NSFC; contract grant numbers: 11301503 (to X.Z.).
The Order of Automorphisms of Quasigroups
Version of Record online: 13 FEB 2014
© 2014 Wiley Periodicals, Inc.
Journal of Combinatorial Designs
Volume 23, Issue 7, pages 275–288, July 2015
How to Cite
McKay, B. D., Wanless, I. M. and Zhang, X. (2015), The Order of Automorphisms of Quasigroups. J. Combin. Designs, 23: 275–288. doi: 10.1002/jcd.21389
- Issue online: 6 MAY 2015
- Version of Record online: 13 FEB 2014
- Manuscript Revised: 3 JAN 2014
- Manuscript Received: 8 OCT 2013
- Australian Research Council. Grant Numbers: DP1093320, FT110100065
- NSFC. Grant Number: 11301503
- Latin square;
- Steiner triple system;
- 1-factorization MSC Classification: 05B15;
- 20N05 (05B05;
We prove quadratic upper bounds on the order of any autotopism of a quasigroup or Latin square, and hence also on the order of any automorphism of a Steiner triple system or 1-factorization of a complete graph. A corollary is that a permutation σ chosen uniformly at random from the symmetric group will almost surely not be an automorphism of a Steiner triple system of order n, a quasigroup of order n or a 1-factorization of the complete graph . Nor will σ be one component of an autotopism for any Latin square of order n. For groups of order n it is known that automorphisms must have order less than n, but we show that quasigroups of order n can have automorphisms of order greater than n. The smallest such quasigroup has order 7034. We also show that quasigroups of prime order can possess autotopisms that consist of three permutations with different cycle structures. Our results answer three questions originally posed by D. Stones.