Block (permutation group theory)

In mathematics and group theory, a block for the action (mathematics)|action] of a group on a set is a subset of whose images under either coincide with or are disjoint from. These images form a block system, a partition of that is -invariant. In terms of the associated equivalence relation on, -invariance means that
for all and all. The action of on induces a natural action of on any block system for.
The set of orbits of the -set is an example of a block system. The corresponding equivalence relation is the smallest -invariant equivalence on such that the induced action on the block system is trivial.
The partition into singleton sets is a block system and if is non-empty then the partition into one set itself is a block system as well. A transitive -set is said to be primitive if it has no other block systems. For a non-empty -set the transitivity requirement in the previous definition is only necessary in the case when and the group action is trivial.

Stabilizers of blocks

If B is a block, the stabilizer of B is the subgroup
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 or the stabilizer G_x is a maximal subgroup of G.