Block (permutation group theory)

This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.
Find sources: "Block" permutation group theory – news · newspapers · books · scholar · JSTOR (June 2019) (Learn how and when to remove this message)

|

Not to be confused with modular representation theory or Aschbacher block.

"Block system" redirects here. For the railway signalling system, see Signalling block system.

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

x ~ y implies gx ~ gy

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

[edit]

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

[edit]

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).

## See also

[edit]

* Congruence relation