A complete group has neither central ambiguity nor outer automorphisms. Embedding it as inner automorphisms creates no larger structure, so it is a fixed point of the automorphism tower. [@isaacs2008]
Definition
Centerlessness gives \(G\cong Inn(G)\), and the second condition is \(Out(G)=1\).
Core result
The inner-automorphism map has kernel \(Z(G)\) and image \(Inn(G)\), proving the first claim. Tower termination uses Wielandt–Schenkman boundedness and finite ascending-chain arguments.
Structural properties
- Every complete group is centerless.
- Having only inner automorphisms is not enough without centerlessness.
- \(S_n\) is complete for most \(n\), with \(S_6\) exceptional because of its outer automorphism.
- A nonabelian simple group need not be complete because it can have outer automorphisms.
- Completeness of a direct product requires care when isomorphic direct factors can be permuted.
Completeness identifies internal conjugation with all external symmetry and is the rigid endpoint of automorphism structure.
Example and boundary
Knowledge network
Automorphism towers terminate at complete groups, symmetric groups provide examples, and generalized Fitting layers help control automorphisms.
Proof and sources
Inner automorphisms, centralizers, and tower stabilization are discussed in Section 9B. [@isaacs2008]
