Near-automorphisms of Latin squares

✍ Scribed by Nicholas J. Cavenagh; Douglas S. Stones

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 560 KB
• Volume: 19
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20282

✦ Synopsis

We define a near-automorphism a of a Latin square L to be an isomorphism such that L and aL differ only within a 2×2 subsquare. We prove that for all n ≥ 2 except n ∈{3, 4}, there exists a Latin square which exhibits a near-automorphism. We also show that if a has the cycle structure (2, n-2), then L exists if and only if n ≡ 2 (mod 4), and can be constructed from a special type of partial orthomorphism. Along the way, we generalize a theorem by Marshall Hall, which states that any Latin rectangle can be extended to a Latin square. We also show that if a has at least 2 fixed points, then L must contain two disjoint non-trivial subsquares.