## Near-automorphisms of Latin squares

##### Citation

Cavenagh, N.J. & Stones, D.S. (2011). Near-automorphisms of Latin squares. Journal of Combinatorial Designs, 19(5), 365-377.

Permanent Research Commons link: http://hdl.handle.net/10289/5155

##### Abstract

We define a near-automorphism α of a Latin square L to be an isomorphism such that L and αL 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 α 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 α has at least 2 fixed points, then L must contain two disjoint non-trivial subsquares.

##### Date

2011##### Type

##### Publisher

Wiley