In mathematics and group theory, a block system for the action of a group G on a set X is a partition of X that is G-invariant. In terms of the associated equivalence relation on X, G-invariance means that

x ~ y implies gx ~ gy

for all g ∈ G and all x, y ∈ X. The action of G on X induces a natural action of G on any block system for X.

The set of orbits of the G-set X is an example of a block system. The corresponding equivalence relation is the smallest G-invariant equivalence on X such that the induced action on the block system is trivial.

The partition into singleton sets is a block system and if X is non-empty then the partition into one set X itself is a block system as well (if X is a singleton set then these two partitions are identical). 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.