Block (permutation group theory)

In mathematics and group theory, a block for the action of a group ${\displaystyle G}$ on a set ${\displaystyle X}$ is a subset ${\displaystyle Y}$ of ${\displaystyle X}$ whose images under ${\displaystyle G}$ either coincide with ${\displaystyle Y}$ or are disjoint from ${\displaystyle Y}$. These images form a block system, a partition of ${\displaystyle X}$ that is ${\displaystyle G}$-invariant. In terms of the associated equivalence relation on ${\displaystyle X}$, ${\displaystyle G}$-invariance means that

${\displaystyle x\sim y}$ implies ${\displaystyle gx\sim gy}$

for all ${\displaystyle g\in G}$ and all ${\displaystyle x,y\in X}$. The action of ${\displaystyle G}$ on ${\displaystyle X}$ induces a natural action of ${\displaystyle G}$ on any block system for ${\displaystyle X}$.¹

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

The partition into singleton sets is a block system and if ${\displaystyle X}$ is non-empty then the partition into one set ${\displaystyle X}$ itself is a block system as well (if ${\displaystyle X}$ is a singleton set then these two partitions are identical). A transitive (and thus non-empty) ${\displaystyle G}$-set ${\displaystyle X}$ is said to be primitive if it has no other block systems.² For a non-empty ${\displaystyle G}$-set ${\displaystyle X}$ the transitivity requirement in the previous definition is only necessary in the case when ${\displaystyle |X|=2}$ and the group action is trivial.

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 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 G_x it's G_B.x = B ∩ G.x and G_H.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 G_x. 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 G_x. 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 G_x is a maximal subgroup of G (then the stabilizers of all elements of X are the maximal subgroups of G conjugate to G_x because G_gx = g ⋅ G_x ⋅ g⁻¹).