A primitive action is a transitive action that cannot be decomposed through a nontrivial block system. It is the irreducible level of permutation-group analysis. [@isaacs2008]
Definition
Primitivity belongs to a particular action, not to the abstract group alone. One group may have both primitive and imprimitive actions.
Core result
Blocks correspond to intermediate subgroups above a point stabilizer, proving maximality. Components of a disconnected orbital graph form blocks, while a nontrivial block produces a disconnected orbital graph.
Structural properties
- Every nontrivial normal subgroup of a primitive group is transitive.
- Doubly transitive actions are primitive.
- Transitive actions of prime degree are primitive.
- An abelian minimal normal subgroup in a primitive group is regular.
- A primitive group containing a transposition is symmetric; one containing a \(3\)-cycle is symmetric or alternating.
Primitive groups are the objects of O'Nan–Scott structure theory, divided into affine, almost simple, and other types through their socles and stabilizer actions.
Example and boundary
Knowledge network
Block systems give the negative condition, orbital graphs give a graph criterion, and multiple transitivity supplies a common sufficient condition.
Proof and sources
Maximal stabilizers, normal orbits, and Jordan-type consequences are in Sections 8B-8C. [@isaacs2008]
