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:

• requiring that the automorphism extend not just once but repeatedly
• replacing extensible automorphism by pushforwardable automorphism, quotient-pullbackable automorphism, extensible endomorphism, or some other closely related notion, and
• replacing automorphism by endomorphism, local isomorphism or some other weaker notion.

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

• Normal-extensible automorphisms problem: This problem seeks to characterize all the normal-extensible automorphisms of a group. A normal-extensible automorphism of a group is an automorphism that can always be extended to a bigger group containing the group as a normal subgroup. Normal-extensible automorphisms of a group need not be inner. In fact, they need not even preserve normal subgroups.
• Characteristic-extensible automorphisms problem: This problem seeks to characterize all the characteristic-extensible automorphisms of a group. These need not be inner.

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.

Extensible automorphisms problems involving order conditions on the group

• Hall-semidirectly extensible implies inner: A Hall-semidirectly extensible automorphism is an automorphism of a finite group that can be extended to any bigger group containing the given subgroup as a Hall subgroup with a normal complement (i.e., as a Hall retract). It turns out that any such automorphism is inner.

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

Further information: Interpretation of the extensible automorphisms problem using 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

Further information: Using group actions and representations to solve the extensible automorphisms problem, Conjugacy class-representation duality

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.