An automorphism tower identifies a centerless group with its inner automorphism group and repeatedly adjoins all automorphisms. Wielandt's theorem says that a finite tower cannot produce essentially new groups forever. [@isaacs2008]
Definition
Centerlessness makes each inner-automorphism map injective. Groups with center require a more general tower construction.
Core result
The inner automorphism group is normal and self-centralizing in the full automorphism group. Schenkman's theorem controls the perfect residual of self-centralizing subnormal subgroups, and Wielandt's boundedness prevents indefinite order growth, forcing the finite chain to stabilize.
Structural properties
- Stability is equivalent to the current group being complete.
- Each group embeds normally as the inner automorphism group of the next.
- Centralizer conditions prevent new independent normal structure from accumulating.
- The generalized Fitting subgroup supplies a canonical layer controlling automorphism action.
- For infinite groups, towers can require transfinite stages and behave more subtly.
The theorem shows that repeatedly adjoining outer automorphisms is a finite process for finite centerless groups.
Example and boundary
Knowledge network
Complete groups are fixed points, generalized Fitting layers control the interior, and self-centralizing subnormal theorems provide bounds.
Proof and sources
Wielandt's tower theorem and the Schenkman–Pettet proof route are Theorems 9.10-9.22. [@isaacs2008]
