A group action G×Ω->Ω might preserve a special kind of partition of Ω called a system of blocks. A block is a subset Δ of Ω such that for any group element g either 1.g preserves Δ, i.e., g Δ = Δ, or 2.g translates everything in Δ out of Δ, i.e., g Δ intersection Δ = ϕ. For example, the general linear group GL(2, R) acts on the plane minus the origin, R^2 - (0, 0). The lines A = {(a t, b t)} are blocks because either a line is mapped to itself, or to another line. Of course, the points on the line may be rescaled, so the lines in A are minimal blocks.

fundamental domain | group | primitive group action | Steiner system