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Finite-extensible implies class-preserving

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Any [[finite-extensible automorphism]] of a [[finite group]] is [[class-preserving automorphism]].
 
==Related facts==
 
===Other facts about finite groups proved using the same method===
 
* [[Finite solvable-extensible implies class-preserving]]: Essentially, the same proof works, because if the original group is solvable, all the bigger groups constructed are also solvable.
* [[Finite-quotient-pullbackable implies class-preserving]]
* [[Hall-extensible implies class-preserving]]
 
===Facts about infinite groups proved using similar constructions===
 
* [[Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving]]
* [[Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving]]
 
===Other results towards the associated conjecture/problem===
 
{{further|[[Extensible automorphisms problem]], [[extensible automorphisms conjecture]], [[finite-extensible automorphisms conjecture]]}}
 
This fact is part of an attempt to prove the [[finite-extensible automorphisms conjecture]], which states that every [[finite-extensible automorphism]] of a finite group must be an [[inner automorphism]]. The finite-extensible automorphisms conjecture is closely related to the [[extensible automorphisms conjecture]], which makes a similar statement about [[extensible automorphism]]s of (possibly infinite) groups. Some related results:
 
* [[Extensible implies subgroup-conjugating]]
* [[Finite-extensible implies subgroup-conjugating]]
==Facts used==
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