# Finite-extensible implies subgroup-conjugating

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., subgroup-conjugating automorphism)

View all automorphism property implications | View all automorphism property non-implications

Get more facts about finite-extensible automorphism|Get more facts about subgroup-conjugating automorphism

## Statement

Suppose is a finite group and is a finite-extensible automorphism of . In other words, for any finite group containing , there is an automorphism of whose restriction to equals .

Then, is a subgroup-conjugating automorphism of : it sends every subgroup of to a conjugate subgroup.

This is a partial result towards the finite-extensible automorphisms conjecture.

## Related facts

- Extensible implies subgroup-conjugating: Essentially, the same proof idea works.
- Finite-extensible implies class-preserving