Extensible equals inner

Statement
The following are equivalent for an automorphism of a group:


 * 1) The automorphism is an extensible automorphism i.e., it can be extended to an automorphism for any group containing it.
 * 2) The automorphism is an inner automorphism.

Extensible automorphism
An automorphism of a group is said to be extensible if it can be extended to an automorphism for every embedding of the group in a bigger group.

In symbols, an automorphism $$\sigma$$ of a group $$G$$ is said to be extensible if for any group $$H$$ containing $$G$$, there is an automorphism $$\sigma'$$ of $$H$$ such that the restriction of $$\sigma'$$ to $$G$$ is $$\sigma$$.

Inner automorphism
An automorphism of a group is said to be inner if it can be expressed as conjugation by some element of the group.

In symbols, an automorphism $$\sigma$$ of a group $$G$$ is said to be inner if there exists $$g \in G$$ such that $$\sigma = c_g$$, i.e., is the map $$x \mapsto gxg^{-1}$$.

Other related facts

 * Finite-extensible implies inner
 * Finite-quotient-pullbackable implies inner

Facts used

 * 1) uses::Inner implies extensible
 * 2) uses::Every group is a malnormal subgroup of a complete group

Proof
Note that because of fact (1) (which is very easy to prove), it suffices to prove only the other direction: that every extensible automorphism of a group is inner.

Schupp's proof
This proof uses fact (2).

Inner implies extensible
Every inner automorphism of a group is extensible. In fact, if an inner automorphism occurs as conjugation by a particular element, it can be extended to conjugation by the same element in any bigger group. This means that inner automorphisms are not just extensible, they are also infinity-extensible (i.e., they can be extended infinitely often) and are also diagram-extensible: they can be extended in a consistent manner for all the members of any lattice of subgroups of a big group, all containing that subgroup.

Variety interpretation
To understand this justification, we need to consider a slightly more general setup. Let $$\mathcal{V}$$ be a variety of algebras. Roughly, there is a collection of operations of fixed arities, and some universally quantified identities for these operations. An algebra in $$\mathcal{V}$$ is a set with those operations satisfying those universal identities.

A variety-extensible automorphism is an automorphism of an algebra in $$\mathcal{V}$$ that extends to an automorphism for any algebra in $$\mathcal{V}$$ containing it.

There is a general notion of I-automorphism of a variety. An I-automorphism of a variety is an automorphism that can be described by a formula (in terms of the operations, and in terms of parameters taking values in that algebra) such that that formula always gives an automorphism for any algebra of the variety and any values assigned to the parameters. I-automorphisms are clearly extensible; in fact, they are infinity-extensible and diagram-extensible. Moreover, they are the only automorphisms whose very form makes it clear that they are extensible.

In the variety of groups, the I-automorphisms are precisely the inner automorphisms. The extensible automorphisms conjecture thus states that in the variety of groups, the only variety-extensible automorphisms are the I-automorphisms. This is a statement about the structural rigidity of the variety of groups. Note that there are many varieties for which there do exist variety-extensible automorphisms that are not I-automorphisms. For more on this, refer interpretation of the extensible automorphisms problem using model theory and universal algebra.