If the action of an automorphism group of a symmetric design on the block set is known, then the design can be constructed by finding one (base) block from each block orbit and then applying the automorphism group to obtain the remaining blocks. If a symmetric design admits an automorphism group such that all blocks of the design form a single orbit, then the group itself can be regarded as the point set of the design. The base block becomes a subset of the group and such subsets are called difference sets. The designs obtained from difference sets admit group invariant incidence matrices.

Group rings are a natural setting for investigating difference sets. The notion of a group of symmetries of a subset of a group ring will be crucial to constructing symmetric designs in subsequent chapters.

Group invariant matrices and group rings

A group invariant matrix is a matrix of order v whose columns can be obtained from the first column by applying all elements of a certain permutation group of order v to the entries of the first column. If we assume that the rows and columns of the matrix are indexed by the elements of the group, then this description leads us to the following definition.