Cubes of symmetric designs and group actions

A 3-dimensional cube of a symmetric (v,k,λ ) ( v , k , λ ) design is a map from [v]³ [ v ] 3 to \0,1\ { 0 , 1 } such that every slice of a cube is an incidence matrix of a (v,k,λ ) ( v , k , λ ) symmetric design. We introduce a regular action of a group G of order v on slices of a cube and generalize classical results about regular action of a group on a symmetric design to a case of dimension 3. Additionally, we prove that a group action on one set of parallel slices fully determines an action of G on other perpendicular slices. We also prove that it is not possible to have two sides with a common first coordinate as left (or right) G -orbits simultaneously. Furthermore, we prove that possible G orbits of mutually perpendicular slices alternate in a sense of being a left and right G -orbit. These results are extended on difference cubes as well.