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
for all "g" ∈ "G" and all "x", "y" ∈ "X". The action of "G" on "X" induces a natural action of "G" on any block system for "X".
The set of orbits of the "G"-set "X" is an example of a block system. The corresponding equivalence relation is the smallest "G"-invariant equivalence on "X" such that the induced action on the block system is trivial.
The partition into singleton sets is a block system and if "X" is non-empty then the partition into one set "X" itself is a block system as well (if "X" is a singleton set then these two partitions are identical). A transitive (and thus non-empty) "G"-set "X" is said to be primitive if it has no other block systems. For a non-empty "G"-set "X" the transitivity requirement in the previous definition is only necessary in the case when |"X"|="2" and the group 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
"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"−"1""x", and by the same method it follows that "g"−"1""B" ⊆ "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
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").