Induction theorems for generalized Bhaskar Rao designs.

Adrian M. Nelson

Abstract

There are extensive results known for the existence of generalized Bhaskar Rao designs signed over solvable groups, and particularly for designs with block size \(3\). There have so far been no comparable results for any non-solvable groups and in particular none for the non-solvable group of smallest order, the simple group \(\mathbb{A}_5\). In this paper we define the new notion of pairwise balanced signed block designs, signed over a group. Our central new result is then a composition theorem for these pairwise balanced signed block designs. From this we derive a pair of induction theorems specifically for constructing generalized Bhaskar Rao design pieces. These induction theorems give conditions under which generalized Bhaskar Rao designs pieces signed over a group can be induced from such designs signed over a subgroup. This is in contrast to long established results which give conditions under which generalized Bhaskar Rao design pieces signed over a quotient group can be inflated to give such designs signed over the whole group. By making systematic use of our new induction theorems and various piecewise constructions we are able to elegantly establish that the well known necessary condition for the existence of generalized Bhaskar Rao designs of block size \(3\) are also sufficient for designs signed over the non-solvable groups \(\mathbb{A}_5\), \(\mathbb{Z}_2\times\mathbb{A}_5\) and \(\mathbb{S}_5\). In the course of these applications we identify a number of new generic generalized Bhaskar Rao design pieces. Finally, and independently of the new induction theorems, we identify a new infinite family of solvable groups for which the known necessary conditions for the existence of generalized Bhaskar Rao designs of block size \(3\) are also sufficient.

Keywords: Generalized Bhaskar Rao designs, difference matrices, group divisible designs, holey generalized Bhaskar Rao designs.

AMS Subject Classification: Primary 05B05; secondary 05B10, 05B30, 51E05.