# Extensible implies permutation-extensible

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

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## Contents

## Statement

Any extensible automorphism of a group is a permutation-extensible automorphism.

## Definitions used

### Extensible automorphism

`Further information: Extensible automorphism`

An automorphism of a group is termed **extensible** if, for any embedding of in a bigger group , there exists an automorphism of such that the restriction of to equals .

### Permutation-extensible automorphism

`Further information: Permutation-extensible automorphism`

An automorphism of a group is termed **permutation-extensible** if, for any embedding of in a symmetric group , there exists an element such that if is conjugation by , the restriction of to is . In other words, extends to an inner automorphism of .

## Related facts

### Applications

- Extensible implies subgroup-conjugating
- Extensible implies normal
- Extensible automorphism-invariant equals normal

## Facts used

- Symmetric groups on finite sets are complete: For a natural number other than or , the symmetric group on elements is a complete group. In particular, every automorphism of it is inner.
- Symmetric groups on infinite sets are complete: The symmetric group on any infinite set is a complete group. In particular, every automorphism of it is inner.

## Proof

**Given**: A group , an extensible automorphism of . A set with an embedding .

**To prove**: extends to an inner automorphism of .

**Proof**: We consider the following cases:

- is infinite: By assumption, extends to an automorphism of . By fact (2), this automorphism must be inner. Hence, extends to an inner automorphism of .
- is finite, and its cardinality is different from or : By assumption, extends to an automorphism of . By fact (2), this automorphism must be inner. Hence, extends to an inner automorphism of .
- is finite with cardinality : By assumption, extends to an automorphism of . But there's only one automorphism of the symmetric group on a two-element set: the identity automorphism. This is clearly inner, so we are done.
- is finite with cardinality : We consider two cases.
- There is an element such that every element of fixes : In this case, is a subgroup of the subgroup , which is the symmetric group on a set of size five. Since is extensible, it extends to an automorphism of , and by fact (1), this automorphism must be inner. This inner automorphism can further be extended to an inner automorphism of , by using the same permutation.
- There is no element of fixed by all elements of : Let with acting on trivially. Thus, acts on , with . is a set of size seven. Since is extensible, it extends to an automorphism of , and by fact (1), this automorphism must be inner. Suppose is a permutation giving this inner automorphism. Then, since fixes , fixes . Since , we get that fixes . Since no element of is fixed by the whole of , . Thus, the permutation restricts to a permutation on the subset , and this inner automorphism gives the required inner automorphism extending to .