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

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(New page: ==Statement== Suppose <math>G</math> is a fact about::centerless group. Consider the automorphism tower of <math>G</math>, def...)
 
(Related facts)
 
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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-Abelian implies automorphism group is complete]]
+
* [[Characteristically simple and non-abelian implies automorphism group is complete]]

Latest revision as of 01:35, 6 March 2013

Statement

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

Then, there exists a natural number n such that G_n is a complete group: the map from G_n to G_{n+1} 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 G_is form an ascending chain of subgroups.

Related facts