Premium
The Order of Automorphisms of Quasigroups
Author(s) -
McKay Brendan D.,
Wanless Ian M.,
Zhang Xiande
Publication year - 2015
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21389
Subject(s) - quasigroup , mathematics , automorphism , combinatorics , order (exchange) , permutation (music) , steiner system , permutation group , factorization , corollary , group (periodic table) , discrete mathematics , algorithm , chemistry , physics , organic chemistry , finance , acoustics , economics
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 S n 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 K n . 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.