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Block (permutation group theory)
In mathematics and group theory, a block for the action of a group G on a set X is a subset of X whose images under G either coincide with X or are disjoint from X. These images form a block system, a partition of X that is G-invariant. In terms of the associated equivalence relation on X, G-invariance means that
for all g\in G and all x,y\in X. The action of G on X induces a natural action of G on any block system for X.{{citation
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.{{citation
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 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).
References
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