x ≡ y implies gx ≡ gy

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

Each element of the block system is called a block. A block can be characterized as a subset B of X such that for all g in G, either

• gB = B (g fixes B) or
• gB ∩ B = ∅ (g moves B entirely).

If B is a block then gB is a block for any g in G. If G acts transitively on X, then the set {gB | g ∈ G} is a block system on X.

The trivial partitions into singleton sets and the partition into one set X itself are block systems. A transitive G-set X is said to be primitive if contains no nontrivial partitions.

See also

• Primitive permutation group
• Congruence relation