Sequences in Dihedral Groups with Distinct Partial Products

Abstract

Given a subset S of the non-identity elements of the dihedral group of order 2m, is it possible to order the elements of S so that the partial products are distinct? This is equivalent to the sequenceability of the group when |S| = 2m − 1 and so it is known that the answer is yes in this case if and only if m > 4. We show that the answer is yes when |S| ≤ 9 and m is an odd prime other than 3, when |S| = 2m − 2 and m is even or prime, and when |S| = 2m − 2 for many instances of the problem when m is odd and composite. We also consider the problem in the more general setting of arbitrary non-abelian groups and discuss connections between this work and the concept of strong sequenceability.