Cycle structures with more than three nontrivial cycles of automorphisms of Latin squares  

A permutation α ∈ Sn is said to be an automorphism of a Latin square if the Latin square is mapped to itself by permuting rows, columns and symbols by α. Let Aut(n) be the set of all automorphisms of Latin squares of order n. Whether a permutation α belongs to Aut(n) depends only on the cycle structure of α. Stones et al characterised α∈ Aut(n) for which α has at most three nontrivial cycles. In this research we prove some related results for automorphisms with four non trivial cycles. Also we found a necessary and sufficient condition for automorphisms opposite to above results when α has finite number of nontrivial cycles such that each cycle is divisible by the next.