# Difference between revisions of "Wielandt's automorphism tower theorem"

From Groupprops

(New page: ==Statement== Suppose <math>G</math> is a fact about::centerless group. Consider the automorphism tower of <math>G</math>, def...) |
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* [[Centerless implies inner automorphism group is centralizer-free in automorphism group]] | * [[Centerless implies inner automorphism group is centralizer-free in automorphism group]] | ||

* [[Centerless and characteristic in automorphism group implies automorphism group is complete]] | * [[Centerless and characteristic in automorphism group implies automorphism group is complete]] | ||

− | * [[Characteristically simple and non- | + | * [[Characteristically simple and non-abelian implies automorphism group is complete]] |

## Latest revision as of 01:35, 6 March 2013

## Statement

Suppose is a Centerless group (?). Consider the automorphism tower of , defined as follows:

- .
- is the Automorphism group (?) of , with the homomorphism from to being the one arising by the natural conjugation action.

Then, there exists a natural number such that is a complete group: the map from to is an isomorphism.

Note that for a centerless group, the inner automorphism group is centralizer-free in the automorphism group. Thus, the automorphism group is again centerless. Further, since we know that the natural map from a centerless group to its automorphism group is injective (it identifies the group with its inner automorphisms), we obtain that the s form an ascending chain of subgroups.