Extensible equals inner

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., inner automorphism)
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Statement

Any extensible automorphism of a group is an inner automorphism.

Definitions

Extensible automorphism

Further information: 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 $G$ such that the restriction of $\sigma '$ to $G$ is $\sigma$.

Inner automorphism

Further information: 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 $\sigma =c_{g}$.

Related facts

Converse

• Inner implies extensible

Other related facts

• Finite-extensible implies inner
• Finite-quotient-pullbackable implies inner

Justification/appeal of the conjecture

Inner implies extensible

For full proof, refer: Inner implies extensible, Inner implies infinity-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

Further information: I-automorphism implies variety-extensible automorphism, Inner automorphisms are I-automorphisms in variety of groups, Interpretation of the extensible automorphisms problem using universal algebra and model theory

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.

References

Journal references

• A characterization of inner automorphisms by Paul E. Schupp, Proceedings of the American Mathematical Society, Volume 101,Number 2, Page 226 - 228(October 1987): ^{JSTOR linkMore info}