The Order of Automorphisms of Quasigroups. Issue 7 (13th February 2014)

• Record Type: Journal Article
• Authors: McKay, Brendan D.; Wanless, Ian M.; Zhang, Xiande
• Is Part Of: Journal of combinatorial designs. Volume 23:Issue 7(2015:Jul.)
• Page Start: 275
• Page End: 288
• Publication Date: 2014-02-13
• Subjects: Combinatorial designs and configurations -- Periodicals; Configurations et schémas combinatoires -- Périodiques; 511.6
• DOI: 10.1002/jcd.21389
• Languages: English
• ISSNs: 1063-8539

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>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 <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjkt8g34p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21389:jcd21389-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="script">S</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></alternatives></inline-formula> will almost surely not be an automorphism of a Steiner triple system of order <italic>n</italic>, a quasigroup of order <italic>n</italic> or a 1‐factorization of the complete graph <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjkt8g335" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21389:jcd21389-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></alternatives></inline-formula>. Nor will σ be one component of an autotopism for any Latin square of order <italic>n</italic>. For groups of order <italic>n</italic> it is known that automorphisms must have order less than <italic>n</italic>, but we show that quasigroups of order <italic>n</italic> can have automorphisms of order greater than <italic>n</italic>. 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.</p> </abstract>