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

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The extensible automorphisms problem for groups and for varieties of algebras

The extensible automorphisms problem and variants for groups

An automorphism $\sigma$ of a group $G$ is termed an 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 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 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, and quotient-pullbackable. It turns out that the converses of all of these are true: extensible equals inner (from which it also follows that pushforwardable implies inner) and quotient-pullbackable implies inner. However, the proof requires non-obvious group constructions.

This page does not discuss the proof of the statement, but rather discusses its significance, why we might expect it to be true, and what its truth says about the theory of groups.

Iterative variants of these notions

Further information: Iteratively extensible automorphism, infinity-extensible automorphism

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.

I-automorphisms and inner automorphisms

The notion of I-automorphism

Further information: 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

Further information: Inner automorphisms are I-automorphisms in the variety of 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 $\infty$-extensible. Similarly, they are $\infty$-pushforwardable and $\infty$-quotient-pullbackable.

Variety-extensible automorphisms need not be I-automorphisms

The statement that extensible equals inner 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 pushforwardable and quotient-pullbackable automorphisms.

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

Further information: Injective module implies every automorphism is infinity-extensible, Injective module implies every endomorphism is infinity-extensible, Abelian-extensible automorphism not implies power map

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.

Automorphisms and endomorphisms

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

Further information: Extensible automorphisms problem

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.