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

x\sim y

implies

gx\sim gy

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 $X$ .^[1]

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.^[2] 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.

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

^ 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
^ 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

Stabilizers of blocks

References

See also