A block is a set of points that the permutation group cannot partially break apart. The translates of a nontrivial block form a systematic partition revealing imprimitive structure. [@isaacs2008]
Definition
Singletons and all of \(\Omega\) are trivial blocks; all others are nontrivial.
Core result
Two translates of an intermediate-subgroup orbit that meet are equal. Conversely, the setwise stabilizer of a block is transitive on that block. The translates partition \(\Omega\), proving divisibility.
Structural properties
- A nontrivial block exists exactly when the action is imprimitive.
- Orbits of a normal subgroup are always blocks.
- The induced action on blocks gives a lower-degree quotient representation.
- Actions within and between blocks often embed in a wreath product.
- A transitive action of prime degree has no nontrivial blocks.
Blocks divide a large permutation problem into action between blocks and action inside one block, providing a recursive interface.
Example and boundary
Knowledge network
Primitive actions exclude nontrivial blocks, the permutation core article gives the subgroup correspondence, and wreath products are the standard container.
Proof and sources
Block partitions, intermediate-subgroup correspondence, and normal orbits are Theorems 8.11-8.15. [@isaacs2008]
