# Extensible automorphisms problem

This article describes an open problem in the following area of/related to group theory: group theory

## Statement

Suppose $\alpha$ is a group property. The extensible automorphisms problem for $\alpha$ is the problem of determining all the group property-conditionally extensible automorphisms with respect to property $\alpha$. An automorphism $\sigma$ of a group $G$ satisfying $\alpha$ is extensible conditional to $\alpha$ if for any group $H$ containing $G$ and satisfying $\alpha$, $\sigma$ extends to an automorphism $\sigma'$ of $H$.

Variants involve:

## The three main formulations and their resolutions for different group properties

### Extensible, pushforwardable, and quotient-pullbackable automorphisms

Suppose $\alpha$ is a group property, $G$ a group satisfying $\alpha$, and $\sigma$ an automorphism of $G$. We say that $\sigma$ is:

• extensible with respect to $\alpha$ if for any group $H$ containing $G$ and satisfying $\alpha$, there is an automorphism $\sigma'$ of $H$ whose restriction to $G$ equals $\sigma$.
• pushforwardable with respect to $\alpha$ if for any group $H$ satisfying $\alpha$ and homomorphism $\rho:G \to H$, there exists an automorphism $\sigma'$ of $H$ such that $\sigma'\circ \rho = \rho \circ \sigma$.
• quotient-pullbackable with respect to $\alpha$ if for any group $K$ satisfying $\alpha$ and surjective homomorphism $\rho:K \to G$, there exists an automorphism $\sigma'$ of $K$ such that $\rho \circ \sigma' = \sigma \circ \rho$.

An inner automorphism of a group satisfies all these properties.

### Known results

Note that since extensible implies pushforwardable for any property, a yes for extensible implies a yes for pushforwardable.

Group property Extensible equals inner? Pushforwardable equals inner? Quotient-pullbackable equals inner?
any group yes yes yes
finite group yes yes yes
group of prime power order (fixed prime) yes yes yes
solvable group yes yes yes
periodic group yes yes yes
p-group yes yes yes
finite solvable group yes yes yes
pi-group yes yes yes
finite pi-group yes yes yes
abelian group no (see abelian-extensible automorphism) no no (see abelian-quotient-pullbackable automorphism)
finite abelian group no (e.g., inverse map) no no (e.g., inverse map)
group of nilpotency class two  ?  ?  ?

## Variations with conditions on subgroup embeddings

### Replacing automorphisms by other kinds of maps

• Extensible local isomorphisms theorem: This states that any extensible local isomorphism, i.e., any isomorphism between subgroups that can always be extended to an automorphism for any bigger group must in fact extend to an inner automorphism of the given group. We can also consider problems of normal-extensible local isomorphisms and characteristic-extensible local isomorphisms.
• Extensible endomorphisms problem: This problem seeks to classify the extensible endomorphisms, i.e., the endomorphisms of a group that can be extended to endomorphisms for any bigger group containing it. There are also corresponding notions of pushforwardable endomorphism, quotient-pullbackable endomorphism. We can also consider problems of normal-extensible endomorphisms and characteristic-extensible endomorphisms.

### Extensible automorphisms problem on subvarieties of the variety of groups

Further information: Variety-extensible automorphisms problem, Quasivariety-extensible automorphisms problem

Let $\mathcal{V}$ be a variety of algebras and $A$ be an algebra in $\mathcal{V}$. An automorphism $\sigma$ of $A$ is termed $\mathcal{V}$-extensible, or variety-extensible for the variety $\mathcal{V}$, if for any algebra $B$ in $\mathcal{V}$ containing $A$ as a subalgebra, $\sigma$ extends to an automorphism $\sigma'$ of $B$.

We can thus try to characterize the $\mathcal{V}$-extensible automorphisms for various subvarieties $\mathcal{V}$ of the variety of groups. Further, we do not need to restrict ourselves to varieties, and can instead look at the automorphisms extensible for particular quasivarieties.

Also, there are analogous notions of pushforwardability and quotient-pullbackability for automorphisms and endomorphisms for any subvariety of the variety of groups.

Here are some particular problems:

• Fixed-class extensible endomorphisms problem: This problem asks for all the endomorphisms of a nilpotent group of class $c$ that can be extended to endomorphisms for all nilpotent groups of class $c$ containing it. The problem is interesting and nontrivial because there are endomorphisms of this kind that are neither trivial nor automorphisms.

## Interpretations

### Destroying outer automorphisms

Further information: Destroying outer automorphisms

The extensible automorphisms problem, and its many variants, are based on the theme that the only automorphisms of a group that survive passing to bigger groups are the inner ones. In other words, outer automorphisms can be destroyed by passing to bigger groups.

A related result is the NPC theorem: it states that any normal subgroup can be realized as a characteristic subgroup inside some bigger group.

### Universal algebra and model theory

The extensible automorphisms problem, and specifically, the associated fact that extensible automorphisms are inner, can be interpreted as a statement about the nature of the variety of groups in terms of universal algebra, or of the theory of groups in terms of model theory/first-order logic. In these interpretations, we note that inner automorphisms are the only ones given by a formula that is guaranteed to hold for all groups.

### Use of representation-theoretic techniques

Both the use of group actions to prove that extensible automorphisms are subgroup-conjugating and the use of linear representation theory to prove that finite-extensible automorphisms are class-preserving share some common features. While the former is mostly a straightforward application of the fundamental theorem of group actions that establishes a direct correspondence between subgroups and transitive group actions, the latter uses a more subtle conjugacy class-representation duality that allows one to relate linear representations with conjugacy classes.