Extensible automorphisms problem

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

Statement

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

Variants involve:

The three main formulations and their resolutions for different group properties

Extensible, pushforwardable, and quotient-pullbackable automorphisms

Suppose  is a group property,  a group satisfying , and  an automorphism of . We say that  is:

• extensible with respect to  if for any group  containing  and satisfying , there is an automorphism  of  whose restriction to  equals .
• pushforwardable with respect to  if for any group  satisfying  and homomorphism , there exists an automorphism  of  such that .
• quotient-pullbackable with respect to  if for any group  satisfying  and surjective homomorphism , there exists an automorphism  of  such that .

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  be a variety of algebras and  be an algebra in . An automorphism  of  is termed -extensible, or variety-extensible for the variety , if for any algebra  in  containing  as a subalgebra,  extends to an automorphism  of .

We can thus try to characterize the -extensible automorphisms for various subvarieties  of the variety of groups. Further, we do not need to restrict ourselves to varieties, and can instead look at the automorphisms extensible for particlar 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  that can be extended to endomorphisms for all nilpotent groups of class  containing it. The problem is interesting and nontrivial because there are endomorphisms of this kind that are neither trivial nor automorphisms.

Some cases that have been completely resolved:

Extensible automorphisms problems involving order conditions on the group

• Hall-semidirectly extensible implies inner: A Hall-semidirectly extensible automorphism is an automorphism 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

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 linear representation theory to prove that finite-extensible automorphisms are class-preserving and the use of group actions to prove that extensible automorphisms are subgroup-conjugating share some common features. While the latter 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.