Algebraic tiling and splitting groups

A number of problems concerning number theory, tiling Euclidean space, coding theory, and gambling have given rise to problems in group theory, usually involving finite abelian groups.

The terms "tiling" or "tesselation" usually call to mind congruent copies of a convex quadrilateral or of a triangle tiling the plane. They remind us of the herringbone pattern formed by translates of an L-shaped brick in pavement. In these examples, copies of some set fill up another set with no overlaps, except perhaps along common borders.

We shall be concerned with factoring and splitting groups. In these cases, translates of certain subsets of a group will exhaust part or all of a group without overlaps.