Multiple transitivity

\(k\)-transitivity requires the group to carry any ordered configuration of \(k\) distinct points to any other and is much stronger than ordinary transitivity. ^{[@isaacs2008]}

Definition

\(k\)-transitivity implies \((k-1)\)-transitivity; \(k\)-homogeneity is generally weaker.

Core result

Align the first coordinate, then align the remaining coordinates inside its stabilizer. Combining these two steps proves the converse and yields stabilizer-chain order formulas.

Structural properties

• Every doubly transitive action is primitive.
• \(S_n\) is \(n\)-transitive, while \(A_n\) is \((n-2)\)-transitive for \(n\geq4\).
• \(AGL(1,q)\) is doubly transitive on the field.
• With a regular normal subgroup, \(k\)-transitivity of the stabilizer on nonidentity kernel elements yields \((k+1)\)-transitivity of the whole group.
• Finite groups of high transitivity are subject to very strong classification restrictions.

Multiple transitivity recursively embeds local point-stabilizer actions into global action and underlies Jordan theorems and simple-group examples.

Example and boundary

Knowledge network

Primitivity follows at degree two, while symmetric and alternating groups supply the extreme examples.

• Primitive action
• Symmetric group
• Alternating group

Proof and sources

Stabilizer recursion, regular-kernel lifting, and Jordan applications are in Sections 8A-8C. ^{[@isaacs2008]}