The Existence Problem for Strong Complete Mappings of Finite Groups

The Cayley table M and the normal multiplication table N of a finite group G are Latin squares. There exists a Latin square orthogonal to both M and N if and only if G admits strong complete mappings. A natural question to ask is, which finite groups admit strong complete mappings? We will summarize work done on the existence problem for strong complete mappings of finite groups. We will also establish new classes of strongly admissible 2-groups. We will also give theoretical proofs of the strong admissibility of some groups of order 16, whose strong admissibility has only been proved via computer searches.