Inner automorphism to automorphism is right tight for normality

Property-theoretic statement
Consider the subgroup property of being a fact about::normal subgroup. Then, the following function restriction expression for this subgroup property:

Inner automorphism $$\to$$ Automorphism

is a right tight function restriction expression.

Statement with symbols
Suppose $$G$$ is a group and $$\sigma:G \to G$$ is a function. Consider the following property for $$\sigma$$:

There exists a group $$K$$ containing $$G$$ as a normal subgroup, and an inner automorphism $$\alpha$$ of $$K$$ such that the restriction of $$\alpha$$ of $$G$$.

This property is precisely equivalent to $$\sigma$$ being an automorphism. (Note that by definition of normality, any such $$\sigma$$ must be an automorphism, so the content of the theorem is that every automorphism can be realized this way).

Facts used

 * 1) Every group is normal fully normalized in its holomorph: If $$G$$ is a group, there is a group $$\operatorname{Hol}(G)$$ (the holomorph of $$G$$) containing $$G$$ as a normal subgroup, with the property that every automorphism of $$G$$ extends to an inner automorphism in $$\operatorname{Hol}(G)$$.

Other facts using this idea or similar proofs

 * Left transiter of normal is characteristic: If $$H \le K$$ is such that whenever $$K$$ is normal in $$G$$, $$H$$ is also normal in $$G$$, then $$H$$ is characteristic in $$K$$. This fact follows from the statement on this page and the right tightness theorem.
 * Every group is normal fully normalized in its holomorph
 * Normal upper-hook fully normalized implies characteristic

Other related facts
If we drop the requirement that $$G$$ be normal in $$K$$, then the result is no longer true. In fact, we have:


 * Every injective endomorphism arises as the restriction of an inner automorphism
 * Isomorphic iff potentially conjugate: Two subgroups of a group that are isomorphic are conjugate in some bigger group containing it.
 * Isomorphic iff potentially conjugate in finite: Two subgroups of a finite group that are isomorphic are conjugate in some bigger finite group containing it.

Proof using the holomorph
The proof using the holomorph finds a single big group that works for all automorphisms.

Given: A group $$G$$, an automorphism $$\sigma$$ of $$G$$.

To prove: There exists a group $$K$$ containing $$G$$ as a normal subgroup, such that $$\sigma$$ extends to an inner automorphism of $$K$$.

Proof: Let $$K$$ be the holomorph of $$G$$. Fact (1) now completes the proof.

Proof using semidirect product with just that automorphism
This proof uses a more minimalistic construction, but the construction differs for every automorphism. This proof technique generalizes to proving that every injective endomorphism arises as the restriction of an inner automorphism.

Given: A group $$G$$, an automorphism $$\sigma$$ of $$G$$.

To prove: There exists a group $$K$$ containing $$G$$ as a normal subgroup, such that $$\sigma$$ extends to an inner automorphism of $$K$$.

Proof: Consider the group $$K := G \rtimes \mathbb{Z}$$, where the generator of $$\mathbb{Z}$$ acts on $$G$$ by $$\sigma$$. $$G$$ is normal in the semidirect product by definition, and the automorphism $$\sigma$$ extends to the inner automorphism of $$K$$ induced by conjugation by that generator of $$\mathbb{Z}$$.