Orthogonal starters in finite abelian groups

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Two problems are considered. First, the conjecture that all odd abelian groups except Z[subscript 3], Z[subscript 5], Z[subscript 9], and Z[subscript 3] + Z[subscript 3] admit strong starters, is reduced to finding strong starters in five types of groups: the cyclic groups of order 3p, 9p, 3[superscript k] for k > 6, 5 • 3[superscript k] for k > 4, and Z[subscript 3] + Z[subscript 3p] where p is any odd prime greater than five. It is shown that all abelian groups G other than Z[subscript 5] such that three does not divide the order of G admits a strong starter. As well, strong starters are given in some small non-cyclic groups which were previously not known to admit starters. Also, a multiplication theorem for sets of pairwise orthogonal starters is given. An exhaustive computer search for orthogonal starters in odd groups smaller than 19 is carried out. The results require the construction of special permutations of some groups.