Extensible automorphisms problem

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

Statement

An extensible automorphism of a group $G$ is an automorphism $\sigma$ of $G$ such that whenever $G$ is a subgroup of a group $H$, there is an automorphism $\sigma '$ of $H$ whose restriction to $G$ is $\sigma$.

The Extensible automorphisms problem over the variety of groups is as follows: given a group $G$, give a characterization of which automorphisms of $G$ are extensible. In other words, describe the group of extensible automorphisms of $G$. While this formulation of the extensible automorphisms problem has been resolved (see discussion below), other variants remain open.

Variants involve:

• considering automorphisms that are extensible over smaller collections of groups than the whole variety of groups, or restricting to embeddings of a particular kind,
• 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.

These different variants can often be combined, leading to a long list of possible questions.

A basic fact here is that the extensible automorphisms do form a group, and another basic fact is that any inner automorphism of a group is extensible.

Particular forms of the problem

The main problem (resolved)

• Extensible implies inner: The only automorphisms of a group that can be extended to automorphisms for any group containing it are the inner automorphisms.
• Finite-extensible implies inner: This states that the only automorphisms of a finite group that can be extended to automorphisms for any finite group containing it are the inner automorphisms. Some results that are easier to prove are: finite-extensible automorphisms preserve conjugacy classes of elements, and they also preserve conjugacy classes of subgroups.

Other known results are:

• Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving
• Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving

Quotient-pullbackable automorphisms

• Quotient-pullbackable implies inner: This states that any quotient-pullbackable automorphism of a group must be inner. An automorphism $\sigma$ of a group $G$ is quotient-pullbackable if, for any surjective homomorphism $\rho :K\to G$, there exists an automorphism $\sigma '$ of $K$ such that $\rho \circ \sigma '=\sigma \circ \rho$. The best result known for this is the finite case, where it is true that finite-quotient-pullbackable implies inner. An important generalization is: conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving.

Variations where conditions are put on the nature of the subgroup embedding and/or the extension

• 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.
• Semidirectly extensible automorphisms problem: This problem seeks to characterize all the semidirectly extensible automorphisms of a group: all the automorphisms that can be extended to a bigger group where the subgroup has a normal complement, to an automorphism that also preserves the normal complement.

Replacing automorphisms by other kinds of maps

• Extensible local isomorphisms conjecture: The conjecture 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 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:

• Abelian-extensible automorphisms problem, abelian-extensible endomorphisms problem: These problems ask for all the abelian-extensible automorphisms (respectively, abelian-extensible endomorphisms): the automorphisms (respectively endomorphisms) of an abelian group that can be extended to an automorphism (respectively endomorphism) for any abelian group containing it. Analogues for pushforwardable and quotient-pullbackable automorphisms/endomorphisms are of interest, as are variants where we require the supergroup to contain the subgroup as a characteristic subgroup.
• Nilpotent-extensible automorphisms problem: This problem asks for all the nilpotent-extensible automorphisms: automorphisms of a nilpotent group that can be extended to automorphisms for any nilpotent group containing it. In other words, it is the problem of finding the quasivariety-extensible automorphisms for the quasivariety of nilpotent groups. Related problems involve the classification of nilpotent-extensible endomorphisms, nilpotent-quotient-pullbackable automorphisms, etc. Also related:
  • 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.
• Solvable-extensible automorphisms problem: This problem asks for all the solvable-extensible automorphisms: automorphisms of a solvable group that can be extended to automorphisms for any solvable group containing it. In other words, it is the problem of finding the quasivariety-extensible automorphisms for the quasivariety of solvable groups. Analogues for endomorphisms, for pushforwardable and quotient-pullbackable, and for restricted cases where the subgroup is a normal subgroup or characteristic subgroup, are interesting.

Also related:

• p-extensible automorphisms problem: This problem asks for all the automorphisms of a group of prime power order that can be extended to automorphisms for all groups of prime power order containing it. In other words, it is the problem of finding the quasivariety-extensible automorphisms for the quasivariety of finite $p$-groups for fixed $p$.
• p-quotient-pullbackable automorphisms problem: This problem asks for all the automorphisms of a group of prime power order that can be pulled back to automorphisms for all surjective homomorphisms to it from groups of prime power order. The best result known currently is that any such automorphism must itself have prime power order for the same prime. In other words, any $p$-quotient-pullbackable automorphism must be a $p$-automorphism.

Some cases that have been completely resolved:

• Finite solvable-extensible implies inner: Any automorphism of a finite solvable group that extends to any finite solvable group containing it is inner. A weaker result, that is easier to prove, is that finite solvable-extensible implies class-preserving.
• Finite solvable-quotient-pullbackable implies inner. Also related is the weaker, and easier to prove, result that finite solvable-quotient-pulllbackable implies class-preserving.

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.

Multiple iterations

• Iteratively extensible automorphisms problem: This problem asks for all the iteratively extensible automorphisms: automorphisms of a group that can be extended $\alpha$ times, for some ordinal $\alpha$. The extreme version of these are infinity-extensible automorphisms, that are $\alpha$-extensible for every ordinal $\alpha$. No better results are known for iteratively extensible automorphisms than the results already known for extensible automorphisms. Analogous problems are the problem of determining the iteratively pushforwardable automorphisms and iteratively quotient-pullbackable automorphisms.
• Chain-extensible automorphisms problem: This problem asks for all the chain-extensible 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 conjecture in this theme is the NPC conjecture, which states that every normal subgroup is a potentially characteristic subgroup: it is 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 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.