In mathematics and group theory, a block system for the action of a group G on a set X is a G-invariant partition. This means the associated equivalence relation satisfies x ~ y implies gx ~ gy for all g in G. An important example is the set of orbits of X, yielding the smallest such block system with trivial induced action. The partition into singleton sets and the partition into the whole set X are also block systems. A transitive G-set is called primitive if it admits no other block systems, with a special exception when |X|=2 and the action is trivial.
Characterization of blocks
Each element of some block system is called a block. A block can be characterized as a non-empty subset B of X such that for all g ∈ G, either
- gB = B (g fixes B) or
- gB ∩ B = ∅ (g moves B entirely).
Proof: Assume that B is a block, and for some g ∈ G it's gB ∩ B ≠ ∅. Then for some x ∈ B it's gx ~ x. Let y ∈ B, then x ~ y and from the G-invariance it follows that gx ~ gy. Thus y ~ gy and so gB ⊆ B. The condition gx ~ x also implies x ~ g−1x, and by the same method it follows that g−1B ⊆ B, and thus B ⊆ gB. 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.
In particular, if B is a block then gB is a block for any g ∈ G, and if G acts transitively on X then the set {gB | g ∈ G} is a block system on X.
Stabilizers of blocks
If B is a block, the stabilizer of B is the subgroup
GB = { g ∈ G | gB = B }.The stabilizer of a block contains the stabilizer Gx of each of its elements. Conversely, if x ∈ X and H is a subgroup of G containing Gx, then the orbit H.x of x under H is a block contained in the orbit G.x and containing x.
For any x ∈ X, block B containing x and subgroup H ⊆ G containing Gx it's GB.x = B ∩ G.x and GH.x = H.
It follows that the blocks containing x and contained in G.x are in one-to-one correspondence with the subgroups of G containing Gx. In particular, if the G-set X is transitive then the blocks containing x are in one-to-one correspondence with the subgroups of G containing Gx. In this case the G-set X is primitive if and only if either the group action is trivial (then X = {x}) or the stabilizer Gx is a maximal subgroup of G (then the stabilizers of all elements of X are the maximal subgroups of G conjugate to Gx because Ggx = g ⋅ Gx ⋅ g−1).