Destroying outer automorphisms

This is a survey article related to:Extensible automorphisms problem
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This is a survey article related to:Potentially characteristic subgroups characterization problem
View other survey articles about Potentially characteristic subgroups characterization problem

This survey article is about a collection of related (and, for the most part, unsolved) problems on the theme: can outer automorphisms (and notions that depend on them) for a given group be destroyed by passing to bigger groups?

An inner automorphism of a group is an automorphism defined as conjugation by an element, and inner automorphisms of groups are remarkably averse to destruction. For instance, any inner automorphism of a subgroup can be extended to an inner automorphism of the whole group, while any inner automorphism of a quotient can be pulled back to an inner automorphism of the original group. This has to do with the fact that inner automorphisms have witnesses: the particular element used for conjugation, and as long as these witnesses are not destroyed, the automorphism is also intact. In the language of universal algebra, we say that inner automorphisms are I-automorphisms of the variety of groups. (In fact, they are the only ones).

Since outer automorphisms do not enjoy the protection given by witnesses, it may happen that they can be destroyed by passing to suitable supergroups or to groups having the original group as quotient. Even better, we may hope that the new bigger group has no outer automorphisms of its own. A related possibility is that normal subgroups of the original group may become characteristic subgroups in the bigger group.

We discuss conjectures related to each of these possibilities.

The extensible automorphisms problem and conjecture

As the graphic illustrates, a little graffiti can completely destroy the symmetry of the circle. The graffiti idea can, in fact, be used to show that for a number of combinatorial structures, every object can be embedded in a bigger object with no nontrivial symmetries. For instance, every graph can be embedded as a subgraph of a bigger graph with no nontrivial symmetry.

With groups, the situation is different. If $G$ is a subgroup of $H$, and $\sigma$ is an inner automorphism of $G$, $\sigma$ can be extended to an inner automorphism of $H$. In other words, there exists an inner automorphism $\sigma'$ of $H$ such that the restriction of $\sigma'$ to $G$ equals $\sigma$. The proof is essentially the fact that if $g \in G$ is an element for which conjugation by $g$ gives $\sigma$, then conjugation by $g$ in $H$ gives $\sigma'$.

A deeper algebraic explanation of this is that inner automorphisms are given by a formula, and this formula is guaranteed to give an automorphism. In fact, any automorphism arising from a formula guaranteed to give an automorphism must be inner. This general notion is called I-automorphism, and inner automorphisms are I-automorphisms in the variety of groups.

Extensible, pushforwardable, and quotient-pullbackable automorphisms

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 (inner implies pushforwardable) and quotient-pullbackable (inner implies quotient-pullbackable). It turns out that the converses of all these statements hold. Thus, extensible equals inner and quotient-pullbackable equals inner. However, these converses are non-obvious and require clever constructions.

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.

Potentially characteristic subgroups

Definition and a conjecture

Further information: Potentially characteristic equals normal, Finite NPC theorem

A subgroup $H$ of a group $K$ is termed a potentially characteristic subgroup if there exists a group $G$ containing $K$ such that $H$ is a characteristic subgroup of $G$.

Since normality satisfies intermediate subgroup condition, any potentially characteristic subgroup is normal. Obviously, any characteristic subgroup is potentially characteristic, so the property of being potentially characteristic lies somewhere between characteristicity and normality.

It turns out that potentially characteristic equals normal: the property of being potentially characteristic coincides with the property of being normal. This is not obvious and requires a clever algebraic construction. Not only do they coincide when considered for all groups, it is also true if we restrict all groups under consideration to be finite.

Potentially relatively characteristic subgroup

A subgroup $H$ of a group $K$ is termed a potentially relatively characteristic subgroup if there exists a group $G$ containing $K$ such that every automorphism of $G$ that restricts to an automorphism of $K$ also restricts to an automorphism of $H$.

It turns out that being potentially relatively characteristic is equivalent to being normal (this is weaker than the statement that potentially characteristic equals normal, but it is easier to prove). Note that this proof is specific to the variety of groups and the analogous statement does not hold when we restrict attention to subvarieties of the variety of groups.

Characteristic-/normal-potentially (relatively) characteristic

A subgroup $H$ of a group $K$ is termed a characteristic-potentially characteristic subgroup if there exists a group $G$ containing $K$ such that both $H$ and $K$ are characteristic in $G$.

A subgroup $H$ of a group $K$ is termed a normal-potentially characteristic subgroup if there exists a group $G$ containing $K$ such that both $H$ is characteristic in $G$ and $K$ is normal in $G$.

A subgroup $H$ of a group $K$ is termed a normal-potentially relatively characteristic subgroup if there exists a group $G$ containing $K$ such that $K$ is normal in $G$ and $H$ is invariant under all automorphisms of $G$ that restrict to automorphisms of $K$.

The implication chain is:

Characteristic-potentially characteristic $\implies$ Normal-potentially characteristic $\implies$ Normal-potentially relatively characteristic

(Note that characteristic-potentially relatively characteristic collapses to characteristic-potentially characteristic).

We might try to make the ambitious conjecture that every normal subgroup is normal-potentially relatively characteristic. This, however, is not true.

Normal-extensible automorphisms

A normal-extensible automorphism is an automorphism that can be extended for any embedding as a normal subgroup of a bigger group. A subgroup of a group is termed a normal-extensible automorphism-invariant subgroup if it is invariant under all the normal-extensible automorphisms of the whole group.

It is not true that every normal-extensible automorphism is inner. In fact, a centerless group that is a maximal subgroup of its automorphism group has all its automorphisms normal-extensible, and in particular, it has normal-extensible outer automorphisms. More generally, every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible.

However, not every normal subgroup is invariant under normal-extensible automorphisms. This shows that not every normal subgroup is normal-potentially relatively characteristic. In particular, it shows that not every normal subgroup is normal-potentially characteristic, or characteristic-potentially characteristic.