In mathematics and group theory, a block for the action of a group on a set is a subset of whose images under either coincide with or are disjoint from . These images form a block system, a partition of that is -invariant. In terms of the associated equivalence relation on , -invariance means that
for all and all . The action of on induces a natural action of on any block system for .
The set of orbits of the -set is an example of a block system. The corresponding equivalence relation is the smallest -invariant equivalence on such that the induced action on the block system is trivial.
The partition into singleton sets is a block system and if is non-empty then the partition into one set itself is a block system as well (if is a singleton set then these two partitions are identical). A transitive (and thus non-empty) -set is said to be primitive if it has no other block systems. For a non-empty -set the transitivity requirement in the previous definition is only necessary in the case when and the group action is trivial.
If B is a block, the stabilizer of B is the subgroup
The stabilizer of a block contains the stabilizer G<sub>x</sub> of each of its elements. Conversely, if x â X and H is a subgroup of G containing G<sub>x</sub>, then the orbit H.x of x under H is a block contained in the orbit G.x and containing x.
For any x â X, block B containing x and subgroup H â G containing G<sub>x</sub> it's G<sub>B</sub>.x = B â© G.x and G<sub>H.x</sub> = 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<sub>x</sub>. 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<sub>x</sub>. 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<sub>x</sub> is a maximal subgroup of G (then the stabilizers of all elements of X are the maximal subgroups of G conjugate to G<sub>x</sub> because G<sub>gx</sub> = g â G<sub>x</sub> â g<sup>âÂÂ1</sup>).