Destroying outer automorphisms

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This is a survey article related to:Extensible automorphisms problem
View other survey articles about Extensible automorphisms problem
This is a survey article related to:NPC conjecture
View other survey articles about NPC conjecture

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 be come characteristic subgroups in the bigger group.

We discuss conjectures related to each of these possibilities.

Circlegraffiti.png

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). The conjectures are about the converses of these statements holding:

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.

Best known general results

Further information: Extensible automorphisms problem

The best known general results are that every extensible automorphism sends subgroups to conjugate subgroups, and every automorphism of a finite group that extends to finite groups sends elements to conjugate elements. Many other results are known for variants of the problem that restrict to particular classes of groups or to particular kinds of embeddings.

Potentially characteristic subgroups

Definition and a conjecture

Further information: Potentially characteristic subgroup, NPC conjecture

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.

The NPC conjecture states that every normal subgroup is potentially characteristic.

The NPC conjecture is closely related to the extensible automorphisms problem. While the extensible automorphisms problem says that outer automorphisms can be destroyed by taking bigger groups, the NPC conjecture says that invariance under all automorphisms (which is characteristicity) reduces to invariance under inner automorphisms once we quantify over all bigger groups. We discuss more of the relationship below.

The best known results related to the NPC conjecture are that finite normal subgroups, central subgroups, characteristic subgroups, as well as normal subgroups contained in a member of the upper central series are all potentially characteristic.

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.

The property of being potentially relatively characteristic is somewhere between potentially characteristic and normal. Further, this bears a somewhat more direct relationship with the extensible automorphisms conjecture. For instance, we can see the following:

Strongly/semi-strongly potentially (relatively) characteristic

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

The implication chain is:

Strongly potentially characteristic \implies Semi-strongly potentially characteristic \implies Semi-strongly potentially relatively characteristic

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

We might try to make the ambitious conjecture that every normal subgroup is semi-strongly 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 normal-extensible automorphism-invariant 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.

It turns out that any semi-strongly potentially relatively characteristic subgroup is normal-extensible automorphism-invariant.

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

Single-witness ideas and extreme versions

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