Transitive action

A transitive action can move any point to any other. It is completely encoded by one point stabilizer and the corresponding coset space. ^{[@isaacs2008]}

Definition

After fixing \(\alpha\), the point stabilizer is \(G_\alpha\) and the degree is \(|\Omega|=|G:G_\alpha|\).

Core result

The map \(\alpha^g\mapsto G_\alpha g\) is well-defined and bijective by orbit-stabilizer. The kernel is the intersection of all conjugate point stabilizers.

Structural properties

• All point stabilizers are conjugate.
• The action is faithful exactly when the point stabilizer is core-free.
• Orbits of a normal subgroup form a block system.
• A regular action is a transitive action with trivial stabilizer.
• A primitive action is a transitive action with maximal point stabilizer.

Transitivity compresses global permutation geometry into the embedding problem for one subgroup \(H\leq G\).

Example and boundary

Knowledge network

The permutation-group core article gives the full hierarchy, while regular and primitive actions strengthen stabilizer and block conditions.

• Permutation group
• Regular action
• Primitive action

Proof and sources

The coset model, stabilizers, and regularity criterion are in Section 8A and Theorems 8.1 and 8.5. ^{[@isaacs2008]}