A regular action has a unique group element carrying any chosen point to any other. It identifies the set unambiguously with the group and produces a standard semidirect decomposition. [@isaacs2008]
Definition
In the finite case \(|G|=|\Omega|\). Left and right multiplication of a group on itself are the standard regular actions.
Core result
Transitivity gives \(G=NH\), and regularity gives \(N\cap H=1\). Identifying \(n\) with \(\alpha^n\) directly verifies compatibility of the two \(H\)-actions.
Structural properties
- Every regular action is faithful.
- Every nonidentity element of a regular subgroup is fixed-point-free.
- If \(H\) is transitive on \(N\setminus\{1\}\), then \(N\rtimes H\) is doubly transitive.
- If \(H\) is semiregular there, the semidirect product is Frobenius.
- An abelian minimal normal subgroup of a primitive group is regular.
A regular normal subgroup internalizes the permutation problem: points become kernel elements and the point stabilizer becomes an automorphism group.
Example and boundary
Knowledge network
Transitive actions supply the coset model, Frobenius groups use a semiregular complement, and affine groups are the standard elementary-abelian regular case.
Proof and sources
Regular normal subgroup decomposition and action correspondence are Theorems 8.5-8.8. [@isaacs2008]
