Interpretation of the extensible automorphisms problem using universal algebra and model theory

The extensible automorphisms problem and variants for groups
An automorphism $$\sigma$$ of a group $$G$$ is termed an survey article about::extensible automorphism if whenever $$H$$ is a group containing $$G$$, there is an automorphism $$\sigma'$$ of $$H$$ such that the restriction of $$\sigma'$$ to $$G$$ equals $$\sigma$$.

An automorphism $$\sigma$$ of a group $$G$$ is termed a survey article about::pushforwardable automorphism if whenever $$\rho:G \to H$$ is a homomorphism, there is an automorphism $$\sigma'$$ of $$H$$ such that $$\rho \circ \sigma = \sigma' \circ \rho$$. In other words, $$\sigma$$ can be pushed forward across any homomorphism of groups. Note that pushforwardable automorphisms are extensible, since we can use subgroup inclusions as homomorphisms.

An automorphism $$\sigma$$ of a group $$G$$ is termed a survey article about::quotient-pullbackable automorphism if whenever $$\rho:K \to G$$ is a surjective homomorphism, there is an automorphism $$\sigma'$$ of $$K$$ such that $$\rho \circ \sigma' = \sigma \circ \rho$$. In other words, $$\sigma$$ can be pulled back across any homomorphism of groups.

Inner automorphisms are extensible (inner implies extensible), pushforwardable (inner implies pushforwardable) and quotient-pullbackable (inner implies quotient-pullbackable). The conjectures are about the converses of these statements holding:


 * survey article about::Extensible automorphisms conjecture: Every extensible automorphism of a group is inner.
 * survey article about::Pushforwardable automorphisms conjecture: Every pushforwardable automorphism of a group is inner.
 * survey article about::Quotient-pullbackable automorphisms conjecture: Every quotient-pullbackable automorphism of a group is inner.

The extensible automorphisms problem for varieties/quasivarieties of algebras
Let $$\mathcal{V}$$ be a variety of algebras. A variety of algebras is defined by a collection of fixed arities of operations, and a collection of universally quantified identities these operations must satisfy. An algebra in the variety is a set equipped with operations of these arities, satisfying the universally quantified identities. The variety of groups is an example of a variety of algebras.

Analogous to the definitions of extensible automorphism, pushforwardable automorphism, and quotient-pullbackable automorphism, we have definitions of variety-extensible automorphism, variety-pushforwardable automorphism, and variety-quotient-pullbackable automorphism.

In this article we discuss a general notion of I-automorphism of a variety of algebras that reduces to inner automorphisms for the variety of groups. Further, we describe how the extensible automorphisms conjecture is essentially a statement about the nature and structural rigidity/determinacy of the variety of groups. We discuss other varieties where this structural determinacy fails to hold. We also discuss related problems for particular subvarieties of the variety of groups, and how insights from universal algebra help us.

Note that instead of varieties of algebras, we can also work with a quasivariety of algebras for many purposes, but for simplicity, we restrict the discussion here to varieties.

Iterative variants of these notions
We can define the notion of $$\alpha$$-extensible for any ordinal $$\alpha$$. All automorphisms are $$0$$-extensible, and an automorphism $$\sigma$$ of a group $$G$$ is $$(\alpha + 1)$$-extensible if, for any group $$H$$ containing $$G$$ as a subgroup, there exists an automorphism $$\sigma'$$ of $$H$$ that is $$\alpha$$-extensible, and such that the restriction of $$\sigma'$$ to $$G$$ is $$\sigma$$. For $$\alpha$$ a limit ordinal, an automorphism is $$\alpha$$-extensible if it is $$\gamma$$-extensible for all ordinals $$\gamma < \alpha$$.

An automorphism of a group is termed infinity-extensible if it is $$\alpha$$-extensible for every ordinal $$\alpha$$. We can, analogously, define $$\alpha$$-pushforwardable, $$\infty$$-pushforwardable, $$\alpha$$-quotient-pullbackable, and $$\infty$$-quotient-pullbackable. Further, we can generalize these notions to arbitrary varieties of algebras.

The notion of I-automorphism
Suppose $$\mathcal{V}$$ is a variety of algebras, and $$A$$ is an algebra in $$\mathcal{V}$$. An I-automorphism of $$A$$ is an automorphism that can be expressed as:

$$x \mapsto \varphi(x,u_1,u_2,\dots,u_n)$$

where $$u_1, u_2, \dots, u_n \in A$$ are fixed, and $$\varphi$$ is a word in terms of the operations of the algebra,with the property that for any algebra $$B$$ of $$\mathcal{V}$$, and any choice of values $$v_1,v_2,\dots,v_n \in B$$, the map:

$$x \mapsto \varphi(x,v_1,v_2,\dots,v_n)$$

gives an automorphism of $$B$$.

In other words $$\varphi$$ is guaranteed to give an automorphism.

Inner automorphisms are I-automorphisms
The inner automorphism given by conjugation by $$g$$ is given by the formula:

$$x \mapsto gxg^{-1}$$.

Thus, inner automorphisms are I-automorphisms.

There are no other I-automorphisms for groups
However, if we are looking at subvarieties of the variety of groups, then there may exist other I-automorphisms that work for all groups in that subvariety. For instance, the map sending every element to its inverse is an I-automorphism in the variety of abelian groups.

I-automorphisms are extensible, pushforwardable, and quotient-pullbackable
In fact, I-automorphism can be extended not just once, but infinitely many times. Thus, I-automorphisms are infinity-extensible.

Variety-extensible automorphisms need not be I-automorphisms
The extensible automorphisms conjecture can now be understood as the statement that in the variety of groups, every variety-extensible automorphism is an I-automorphism. Similar interpretations hold for the pushforwardable automorphisms conjecture and the quotient-pullbackable automorphisms conjecture.

In this section, we discuss some other varieties where this conjecture holds, and some others where it does not hold.

The variety of sets
The most extreme example of a variety of algebras is the variety of sets. Here, there are no operations and no identities. Consequently, the only functions that can be written down as words are the identity map and constant maps. From this, we easily see that the identity map is the only I-automorphism in the variety of sets.

On the other hand, every automorphism is extensible in the variety of sets. This is because every permutation of a set can be extended to a permutation of a bigger set containing it, by, for instance, setting the extended permutation to fix every element in the complement. Thus, for sets of size two or more, there are extensible automorphisms that are not I-automorphisms.

Let us now consider the pushforwardable automorphisms on the variety of sets. It turns out that for any set of size more than two, the only pushforwardable automorphism is the identity map. To see this, consider a non-identity pushforwardable automorphism $$\sigma$$ on a set of size at least three. There exist elements $$a \ne b$$ such that $$\sigma(a) = b$$. Consider a map $$S \to \{ 0,1 \}$$ sending $$a$$ to zero and everything else to one. The automorphism $$\sigma$$ cannot be pushed forward along this map. The reason this argument works is that the variety of sets is not congruence-uniform, i.e., given a surjective homomorphism of sets, it is not necessary that the fibers all have equal size.

A similar argument shows that for any set, the only quotient-pullbackable automorphism is the identity map.

Thus, we see that for sets, every automorphism is extensible but the only pushforwardable automorphism (except for a set of size two) is the identity map, and the only quotient-pullbackable automorphism is the identity map. Further, the only I-automorphism is the identity map.

The complete contrast between I-automorphisms and extensible automorphisms shows that the algebraic structure of sets puts very little restriction on the nature of their automorphisms -- this is to be expected since the algebraic structure of sets is empty. On the other hand, the collapse of pushforwardable and quotient-pullbackable automorphisms to the identity map has more to do with the fact that the algebraic structure of sets puts very little restriction on the nature of quotient maps.

The variety of modules
For modules over a commutative unital ring $$R$$, the operations $$m \mapsto rm$$ for any unit $$r \in R$$ are all I-automorphisms, and these are in fact the only ones. Clearly, these automorphisms are extensible, pushforwardable, and quotient-pullbackable.

However, there exist extensible automorphisms of $$R$$-modules that are not I-automorphisms. For instance, let $$M$$ be a nonzero injective $$R$$-module. Consider the module $$M \oplus M$$. This is also injective, and the coordinate exchange automorphism $$(x,y) \mapsto (y,x)$$ is not an I-automorphism. However, it is easy to see that any automorphism of an injective module is extensible. In fact, automorphisms of injective modules are infinity-extensible.

In particular, this results applies to the variety of abelian groups: for instance, we can take $$M = \mathbb{Q}$$ above.

For groups
So far, we have discussed extensibility of automorphisms of a structure. We now discuss a related notion of extensible endomorphism. An extensible endomorphism of a group $$G$$ is an endomorphism $$\sigma$$ of $$G$$ such that for any group $$H$$ containing $$G$$, there is an endomorphism $$\sigma'$$ of $$H$$ such that the restriction of $$\sigma'$$ to $$G$$ is $$\sigma$$.

We can similarly define pushforwardable endomorphism and quotient-pullbackable endomorphism.

Variety interpretation for groups
We can also consider the notion of extensible endomorphism, pushforwardable endomorphism, or quotient-pullbackable endomorphism for an arbitrary variety of algebras. Further, there is a notion of I-endomorphism analogous to the notion of I-automorphism. Any I-endomorphism is extensible, pushforwardable, and quotient-pullbackable. The only I-endomorphisms for the variety of groups are the inner automorphisms and the trivial map (the map sending every element to the identity element).

Thus, extrapolating from the automorphisms situation, we can make the extensible endomorphisms conjecture: the only extensible endomorphisms of a group are the trivial map and inner automorphisms.

Variety interpretation for subvarieties of the variety of groups
For most subvarieties of the variety of groups that we commonly encounter, the I-automorphisms continue to be the inner automorphisms. However, new examples of I-endomorphisms do come up. Here are some examples:


 * For the variety of abelian groups, the map $$x \mapsto nx$$, i.e., the $$n^{th}$$ power map or $$n^{th}$$ multiple map, is an I-endomorphism. In fact, these are the only I-endomorphisms.
 * For the variety of nilpotent groups of class $$c$$, for fixed $$c$$, the map:

$$x \mapsto [[[[\dots [ x,u_1],u_2],\dots,u_{c-2}],u_{c-1}]^r$$

is an I-endomorphism, where the $$u_i$$ are parameters and $$r$$ is a fixed integer. In fact, the image of this endomorphism is inside the center of the group. If the class is strictly less than $$c$$, the map is always trivial. If the class equals $$c$$, there are choices of the parameters $$u_i$$ for which the map is nontrivial. Thus, for the variety of nilpotent groups of class $$c$$, the I-endomorphisms include both the endomorphisms of the above form and the inner automorphisms.

Some associated conjectures and problems
The above observations suggest some initially plausible conjectures for the variety of abelian groups and the variety of nilpotent groups of class $$c$$ for fixed $$c$$. For instance, we might hope that the only extensible endomorphisms of abelian groups are the multiple maps. This is not true, and the counterexample is the same as that used earlier for automorphisms: in the abelian group $$\mathbb{Q} \oplus \mathbb{Q}$$, the coordinate exchange automorphism is not a multiple map. However, since this abelian group is injective as a $$\mathbb{Z}$$-module, every automorphism of it is extensible.

However, the statement is true for finite abelian groups: the only endomorphisms of a finite abelian group that extend to endomorphisms for every finite abelian group containing it are the multiple maps.

The corresponding statement for finite nilpotent groups of class $$c$$ is open.