Block (permutation group theory)

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 for the action of a group ${\displaystyle G}$ on a set ${\displaystyle X}$ is a subset of ${\displaystyle X}$ whose images under ${\displaystyle G}$ either coincide with ${\displaystyle X}$ or are disjoint from ${\displaystyle X}$. 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.

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

1. Seress, Ákos (2003), Permutation Group Algorithms, Cambridge Tracts in Mathematics, vol. 152, Cambridge University Press, Cambridge, p.  9, doi:10.1017/CBO9780511546549, ISBN   0-521-66103-X, MR   1970241
2. Dixon, John D.; Mortimer, Brian (1996), Permutation Groups, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, p.  12, doi:10.1007/978-1-4612-0731-3, ISBN   0-387-94599-7, MR   1409812

See also

• Congruence relation