Cubes of symmetric designs and group actions

Abstract A 3-dimensional cube of a symmetric $$(v,k,\lambda )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> design is a map from $$[v]^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>v</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> to $$\{0,1\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> such that every slice of a cube is an incidence matrix of a $$(v,k,\lambda )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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.