All groups of odd order have starter-translate 2-sequencings

Bailey defined 2-sequencings (terraces) of groups. She conjectured that all finite groups except elementary Abelian 2-groups (other than the cyclic group Z2) have 2-sequencings and proved that the direct product of a 2-sequenceable group and a cyclic group of odd order is 2-sequenceable. It is shown here that all groups of odd order have a special type of 2-sequencing called a starter-translate 2-sequencing. It follows that if G is a group of odd order and the finite group H has a symmetric sequencing, then G x H also has a symmetric sequencing. Bailey used 2-sequencings for the purpose of constructing large numbers of quasi-complete Latin squares of a fixed order. Families of such squares can be useful in certain experiments. With this in mind, an alternative construction for starter-translate 2-sequencings of a large class of groups of odd order is presented. The construction is closely tied to the general method and shows that, with respect to the generation of 2-sequencings, the apparently unimportant step of replacing right cosets of a normal subgroup by left cosets can have major consequences. This method works on a class of groups containing all supersolvable groups of odd order. § 1. INTRODUCTION. In [8] Bailey generalized ideas of Gordon [9] to give an algebraic method for constructing quasi-complete Latin squares. One goal of her work was to find "valid randomization sets" of quasi-complete Latin squares of order n. She was able to do this for odd prime powers. Bailey defined the concept of a terraced group. This idea was rediscovered in [1] where it was called a 2-sequencing and has been the subject of several recent investigations [4,5,7,13]. Bailey conjectured that the only finite groups that cannot be terraced are the elementary Abelian 2-groups Z2 X Z2 X • • • X Z2 with at least two factors. She proved that if B is a terraced group and C = Z2n+l, then B x C is terraced. This was generalized in [5] by allowing C to be any group of odd order that carries a special type of terrace called a starter-translate 2-sequencing. All cyclic groups of odd order satisfy this condition. The first, and major, goal of this paper is to show that its title is a true statement. DEFINITION 1. Suppose G is a group of odd order 2n + 1 and identity e. Then S = {{Xl, YI }, ... ,{x n1 …