Block (permutation group theory)

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 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.

Stabilizers of blocks

If B is a block, the stabilizer of B is the subgroup

G_B = { g ∈ G | gB = B }.

The stabilizer of a block contains the stabilizer G_x of each of its elements. Conversely, if x ∈ X and H is a subgroup of G containing G_x, then the orbit of x under H is a block. It follows that the blocks containing x are in one-to-one correspondence with the subgroups of G containing G_x. In particular, a G-set is primitive if and only if the stabilizer of each point is a maximal subgroup of G.

See also

• Primitive permutation group
• Congruence relation