Varying characteristicity

Characteristicity is a pivotal subgroup property, that, along with normality, dates back to before the twentieth century. The notion was introduced by Frobenius for those subgroups that are fixed by all automorphisms of the group (and not just the inner ones).

This article surveys some of the more common among the variations of the subgroup property of characteristicity. The ideas are:


 * Emulate the strengths
 * Remedy the weaknesses

Intermediately characteristic subgroup
The intermediately operator takes as input a subgroup property and outputs the property of being a subgroup that has the original property in every intermediate subgroup.

Note that a subgroup property is fixed under the intermediately operator if and only if it satisfies the intermediate subgroup condition.

The subgroup property of being characteristic does not satisfy the intermediate subgroup condition. Applying the intermediately operator to it gives the subgroup property of being intermediately characteristic. In other words, $$H$$ is an intermediately characteristic subgroup of $$G$$ if $$H$$ is characteristic in every intermediate subgroup $$K$$.

Some properties that are stronger than being intermediately characteristic include: isomorph-containing subgroup (contains every isomorphic subgroup), isomorph-free subgroup (has no other isomorphic subgroup), homomorph-containing subgroup (contains every homomorphic image), order-unique subgroup, normal Sylow subgroup, and normal Hall subgroup.

Transfer-closed characteristic subgroup
The transfer condition operator takes as input a subgroup property $$p$$ and outputs the property of being a subgroup $$H$$ of $$G$$ such that for any subgroup $$K$$ of $$G$$, $$H \cap K$$ satisfies $$p$$ in $$K$$.

The transfer condition operator applied to the subgroup property of being characteristic is the subgroup property of being transfer-closed characteristic. In other words, $$H$$ is transfer-closed characteristic in $$G$$ if, for every subgroup $$K$$ of $$G$$, $$H \cap K$$ is characteristic in $$K$$.

Some properties that are stronger than being transfer-closed characteristic are: normal Sylow subgroup, normal Hall subgroup, subisomorph-containing subgroup, subhomomorph-containing subgroup, and variety-containing subgroup.

Image-closed characteristic subgroup
The image condition operator takes as input a subgroup property $$p$$ and outputs the property of being a subgroup $$H$$ in a group $$G$$ such that for any surjective homomorphism from $$G$$, the image of $$H$$ satisfies $$p$$ in the image of $$G$$.

Applying the image condition operator to the subgroup property of being characteristic gives the subgroup property of being image-closed characteristic. In other words, a subgroup $$H$$ of a group $$G$$ is termed image-closed characteristic in $$G$$ if, for every surjective homomorphism $$\rho:G \to K$$, $$\rho(H)$$ is a characteristic subgroup of $$K$$.

Potentially characteristic subgroup
The potentially operator is another remedy for a subgroup not satisfying the intermediate subgroup condition. Given a subgroup property $$p$$, the subgroup property potentially $$p$$ is the property of being a subgroup $$H \le K$$ such that there exists a group $$G$$ containing $$K$$ such that $$H$$ satisfies $$p$$ in $$G$$.

Applying the potentially operator to the subgroup property of being characteristic gives the subgroup property of begin a normal subgroup. This is the content of the NPC theorem. See also the potentially characteristic subgroups characterization problem for a general discussion.lly characteristic in a group $$K$$ if there is a group $$G$$ containing $$K$$ such that both $$H$$ and $$K$$ are characteristic in $$G$$.

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

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

It turns out that normal not implies normal-potentially characteristic, and normal not implies normal-potentially relatively characteristic.

The overall implication chain
We have:

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

Invariance properties
Like normality, characteristicity is an invariance property. In other words, it can be written using the function restriction expression:

Automorphism $$\to$$ Function

which can be interpreted in words as: a subgroup $$H$$ of a group $$G$$ is characteristic in $$G$$ if and only if every automorphism of $$G$$ restricts to a well-defined function on $$H$$. Or equivalently, any automorphism of $$G$$ leaves $$H$$ invariant.

Similarly, normality can be expressed as the invariance property:

Inner automorphism $$\to$$ Function

We can choose to vary characteristicity by changing the property on the left side of the formal expression (that is, by taking the invariance property with respect to some other, related function property). Some examples are described below.

Strictly characteristic subgroup
A subgroup of a group is said to be strictly characteristic if every surjective endomorphism of the whole group maps the subgroup to within itself. Strict characteristicity can be expressed as the following invariance property:

Surjective endomorphism $$\to$$ Function

In a finite group, and more generally, in a Hopfian group, every surjective endomorphism is an automorphism. However, for infinite groups, there may be surjective endomorphisms that are not automorphisms. An example is the left-shift operator on the infinite direct power of a group, that simply shifts all coordinates to the left.

Injective endomorphism-invariant subgroup
A subgroup of a group is said to be injective endomorphism-invariant if every injective endomorphism of the whole group maps the subgroup to within itself. This can be expressed as the following invariance property:

Injective endomorphism $$\to$$ Function

In a finite group, and more generally, in a co-Hopfian group, every injective endomorphism is an automorphism. However, for infinite groups, there may be injective endomorphisms that are not automorphisms. An example is the right-shift operator on the infinite direct power of a group, that simply shifts all coordinates to the right.

Fully invariant subgroup
A subgroup of a group is said to be fully invariant or fully characteristic if it is invariant under every endomorphism of the group. This condition is much stronger than characteristicity. Full invariance can be expressed as the following invariance property:

Endomorphism $$\to$$ Function

Normality-preserving endomorphism-invariant subgroup
A normality-preserving endomorphism is an endomorphism that sends normal subgroups to normal subgroups. Note that any automorphism, and more generally, any surjective endomorphism, is normality-preserving, because normality satisfies image condition. A normality-preserving endomorphism-invariant subgroup is a subgroup invariant under all normality-preserving endomorphisms. Any such subgroup is strictly characteristic and also characteristic.

Invariance under nice automorphisms
These definitions are stated here for finite groups, but generalize to arbitrary groups.

A subgroup $$H$$ of a finite group $$G$$ is termed a:


 * cofactorial automorphism-invariant subgroup if it is invariant under all cofactorial automorphisms of $$G$$: automorphisms $$\sigma$$ of $$G$$ with the property that every prime divisor of $$\sigma$$ also divides the order of $$G$$.
 * subgroup-cofactorial automorphism-invariant subgroup if it is invariant under all the automorphisms $$\sigma$$ of $$G$$ with the property that every prime divisor of $$\sigma$$ also divides the order of $$H$$.
 * coprime automorphism-invariant subgroup if it is invariant under all automorphisms whose order is relatively prime to that of $$G$$.

Subgroup-cofactorial automorphism-invariant subgroups behave very similarly to characteristic subgroups in many ways. For instance, subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal.

Isomorph-free and isomorph-containing subgroup
A subgroup $$H$$ of a group $$G$$ is termed isomorph-free in $$G$$ if there is no other subgroup of $$G$$ isomorphic to $$H$$. Isomorph-free subgroups are truly one of their kind, and are in particular, characteristic.

The problem with isomorph-freeness is that it is at times too strong. A group may not be isomorph-free as a subgroup of itself, because it may be isomorphic to a proper subgroup. For instance, the group of integers is isomorphic to proper subgroups of itself.

A somewhat weaker notion is that of an isomorph-containing subgroup. A subgroup $$H$$ of a group $$G$$ is termed isomorph-containing in $$G$$ if any subgroup $$K$$ of $$G$$ isomorphic to $$H$$ is contained in $$H$$.

Other related notions of isomorph-freeness
Here are some other related notions:


 * Quotient-isomorph-free subgroup: A subgroup $$H$$ of a group $$G$$ is termed quotient-isomorph-free if $$H$$ is normal in $$G$$, and such that if $$K$$ is a normal subgroup of $$G$$ such that $$G/K \cong G/H$$, then $$H \le K$$.
 * Series-isomorph-free subgroup: A subgroup $$H$$ of a group $$G$$ is termed series-isomorph-free if $$H$$ is normal in $$G$$, and such that if $$K$$ is normal in $$G$$ with $$H \cong K, G/H \cong G/K$$, then $$H = K$$.
 * Normal-isomorph-free subgroup: A subgroup $$H$$ of a group $$G$$ is termed normal-isomorph-free if $$H$$ is normal in $$G$$, and such that if $$K$$ is a normal subgroup of $$G$$ with $$H \cong K$$, then $$H = K$$.

Some stronger notions
Here are some stronger notions, that make sense particularly for finite subgroups or subgroups of finite index:


 * Order-unique subgroup: A finite subgroup of a group that is the only subgroup with that order.
 * Index-unique subgroup: A subgroup of finite index that is the only subgroup with that index.

When the whole group is finite, these two notions are equivalent.

One of its kind (resemblance)
Characteristic subgroups are those subgroups that are invariant under the group's own symmetries, viz., under the automorphism group of the group. In other words, they are truly invariant, and ideally, we should be able to, for every characteristic subgroup of a group, give a description that is satisfied only by that subgroup and by no other. However, restrictions on the kind of language that we can use force us to come up with different, often stronger, versions of characteristicity for groups that can actually be described.

Elementarily characteristic subgroup
The idea behind introducing elementarily characteristic subgroups is to be able to describe what one would mean by characteristic when one only had first-order logic at one's disposal. An elementarily characteristic subgroup is a subgroup $$H \le G$$ such that there is no other subgroup $$K$$ for which $$H$$ and $$K$$ cannot be distinguished by means of first-order logic. In other words, $$H$$ is the only subgroup of its kind where the kind can be checked for only within first-order logic.

The property of being elementarily characteristic is a fairly strong property.

MSO-characteristic subgroup
A MSO-characteristic subgroup is a subgroup that is unchanged under any equivalence that preserves the structure of the group that can be described using monoidal second-order logic.

Second-order characteristic subgroup
A second-order characteristic subgroup is a subgroup that in unchanged under any equivalence that preserves that structure of the group that can be described using second-order logic.

Purely definable subgroup
This is a subgroup that is a definable subset in the first-order theory of the group (treated as a pure group). In other words, it is the specialization of the notion of definable subgroup, to the case of a pure group.

Second-order purely definable subgroup
This is a subgroup that is a definable subset in the second-order theory of the group (treated as a pure group). In other words, it is the specialization of the notion of second-order definable subgroup, to the case of a pure group.