Finite-quotient-pullbackable implies class-preserving: Difference between revisions

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Suppose <math>G</math> is a [[finite group]] and <math>\sigma</math> is a [[finite-quotient-pullbackable automorphism]] of <math>G</math>. Then, <math>\sigma</math> is a [[class-preserving automorphism]] of <math>G</math>: it sends every element of <math>G</math> to within its [[conjugacy class]].
Suppose <math>G</math> is a [[finite group]] and <math>\sigma</math> is a [[finite-quotient-pullbackable automorphism]] of <math>G</math>. Then, <math>\sigma</math> is a [[class-preserving automorphism]] of <math>G</math>: it sends every element of <math>G</math> to within its [[conjugacy class]].
==Related facts==
* [[Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving]]: We can slightly generalize the proof technique to show that the result holds not just for finite groups but also for [[conjugacy-separable group]]s.
* [[Finite-extensible implies class-preserving]]
* [[Finite-extensible implies subgroup-conjugating]], [[Extensible implies subgroup-conjugating]]
* [[Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving]]


==Facts used==
==Facts used==


# [[uses::Finite-quotient-pullbackable implies Hall-quotient-pullbackable]]
# [[uses::Finite-quotient-pullbackable implies Hall-quotient-pullbackable]]
# [[uses::Hall-quotient-pullbackable implies linearly pushforwardable over prime field]] when the prime does not divide the order of the group.
# [[uses::Hall-quotient-pullbackable implies linearly pushforwardable over prime field]]
# [[uses::Linearly pushforwardable implies class-preserving]] when the field is a [[class-separating field]]
# [[uses::Linearly pushforwardable implies class-preserving]] when the field is a [[class-separating field]]
# [[uses::Every finite group admits a sufficiently large field]]
# [[uses::Every finite group admits a sufficiently large field]]
# [[uses::Sufficiently large implies splitting]], [[uses::Splitting implies character-separating]], [[uses::Character-separating implies class-separating]]
# [[uses::Sufficiently large implies splitting]], [[uses::Splitting implies character-separating]], [[uses::Character-separating implies class-separating]]

Revision as of 18:35, 27 April 2009

This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., finite-quotient-pullbackable automorphism) must also satisfy the second automorphism property (i.e., class-preserving automorphism)
View all automorphism property implications | View all automorphism property non-implications
Get more facts about finite-quotient-pullbackable automorphism|Get more facts about class-preserving automorphism

Statement

Suppose G is a finite group and σ is a finite-quotient-pullbackable automorphism of G. Then, σ is a class-preserving automorphism of G: it sends every element of G to within its conjugacy class.

Related facts

Facts used

  1. Finite-quotient-pullbackable implies Hall-quotient-pullbackable
  2. Hall-quotient-pullbackable implies linearly pushforwardable over prime field
  3. Linearly pushforwardable implies class-preserving when the field is a class-separating field
  4. Every finite group admits a sufficiently large field
  5. Sufficiently large implies splitting, Splitting implies character-separating, Character-separating implies class-separating
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