In terms of the associated equivalence relation on X, G-invariance means that for all g ∈ G and all x, y ∈ X.
The set of orbits of the G-set X is an example of a block system.
A transitive (and thus non-empty) G-set X is said to be primitive if it has no other block systems.
For a non-empty G-set X the transitivity requirement in the previous definition is only necessary in the case when |X|=2 and the group action is trivial.
In the other direction, if the set B satisfies the given condition then the system {gB | g ∈ G} together with the complement of the union of these sets is a block system containing B.