# Finite-extensible implies class-preserving

From Groupprops

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-extensible automorphism) must also satisfy the second automorphism property (i.e., class-preserving automorphism)

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Get more facts about finite-extensible automorphism|Get more facts about class-preserving automorphism

This fact is related to: Extensible automorphisms problem

View other facts related to Extensible automorphisms problemView terms related to Extensible automorphisms problem |

## Statement

Any finite-extensible automorphism of a finite group is class-preserving automorphism.

## Facts used

- Finite-extensible implies finite-characteristic-semidirectly extensible
- Finite-characteristic-semidirectly extensible implies linearly pushforwardable over prime field (where the prime does not divide the order of the group)
- Linearly pushforwardable implies class-preserving when the field is a class-separating field
- Every finite group admits a sufficiently large prime field
- Sufficiently large implies splitting, splitting implies character-separating, character-separating implies class-separating