Metaproperty satisfaction analysis for right transiters of normality

This survey article looks at various subgroup properties $p$ that all behave as follows: any normal subgroup of a subgroup with property $p$ is normal in the whole group.

We discuss how these properties behave with respect to intersections, joins, centralizers, and many other operations.

The properties

The motivation

The motivation behind these properties is to think of normality in terms of a function restriction expression. The standard function restriction expression for normality is:

Inner automorphism $\to$ Automorphism

In other words, $H$ is a normal subgroup of $G$ if any inner automorphism of $G$ restricts to an automorphism of $H$.

However, the left side can be tightened in different ways:

• Subgroup-conjugating automorphism $\to$ Automorphism
• Class-preserving automorphism $\to$ Automorphism
• Normal automorphism $\to$ Automorphism

The properties themselves

The four properties that are of direct interest to us are:

• Central factor: This is defined as inner automorphism $\to$ inner automorphism. In other words, $H$ is a central factor of $G$ if and only if every inner automorphism of $G$ restricts to an inner automorphism of $H$. Equivalently, $HC_G(H) = G$.
• SCAB-subgroup: This is defined as inner automorphism $\to$ subgroup-conjugating automorphism, or equivalently, subgroup-conjugating automorphism $\to$ subgroup-conjugating automorphism. In other words, $H$ is a SCAB-subgroup of $G$ if any inner automorphism of $G$ restricts to an automorphism of $H$ sending subgroups to conjugate subgroups in $H$.
• Conjugacy-closed normal subgroup: This is defined as inner automorphism $\to$ class-preserving automorphism, or equivalently, as class-preserving automorphism $\to$ class-preserving automorphism.
• Transitively normal subgroup: This is defined as normal automorphism $\to$ normal automorphism, or equivalently, as inner automorphism $\to$ normal automorphism. $H$ is transitively normal in $G$ if every normal subgroup of $H$ is normal in $G$.

Some other properties

There are other properties that come up.

Some properties that are stronger than the property of being a central factor:

Some properties that are obtained by stripping the normal from some of the properties earlier:

• Subset-conjugacy-closed subgroup: A subgroup $H$ of a group $G$ is subset-conjugacy-closed in $G$ if, for any subsets $A,B \subseteq H$ such that there exists $g \in G$ with $gAg^{-1} = B$, there exists $h \in H$ such that $hah^{-1} = gag^{-1}$ for all $a \in A$. A subgroup is a central factor if and only if it is both subset-conjugacy-closed and normal.
• Conjugacy-closed subgroup: A subgroup $H$ of a group $G$ is conjugacy-closed if, whenever two elements of $H$ are conjugate in $G$, they are conjugate in $H$. A conjugacy-closed normal subgroup (seen earlier) is a subgroup that is both conjugacy-closed and normal.
• Central factor of normalizer: A subgroup that is a central factor in its normalizer.

Finally, one more property that we shall be frequently looking at is the property of being a retract. A retract is a subgroup that possesses a normal complement; equivalently, it is the image of an idempotent endomorphism of the whole group.

Transitivity

All the properties discussed here are transitive, with the exception of the property of being a central factor of normalizer. The explanations are provided below:

The property of being a central factor of normalizer is not transitive. Further information: Central factor of normalizer is not transitive

Intermediate subgroup condition

All the properties given here satisfy the intermediate subgroup condition. In other words, for any of these properties $p$, if $H \le K \le G$ and $H$ satisfies property $p$ in $G$, then $H$ satisfies property $p$ in $K$.

All the properties described here satisfy the intermediate subgroup condition:

Transfer condition

Most of the properties discussed here do not satisfy the transfer condition. In other words, we can have a situation where $H, K$ are subgroups of $G$ with $H$ satisfying property $p$ in $G$, but $H \cap K$ does not satisfy property $p$ in $K$.

Here is the lone property that satisfies transfer condition:

Here are the many that do not satisfy transfer condition:

Image condition

All of the properties discussed here satisfy the image condition. In other words, the image, under a surjective homomorphism, of a subgroup satisfying any of these properties, also satisfies that property:

Intersection-closed

Most of the properties discussed here are not closed under taking finite intersections. The exception is the property of being a central subgroup, which is closed under taking a finite (nonzero) number of intersections.

Moreover, intersections of subgroups of this kind can be very bad in general.

Join-closed

Most of the properties discussed here are not closed under taking finite joins (the exception being central subgroup and cocentral subgroup):

However, there are interesting relations between the properties. For instance, define a subgroup to be a join-transitively central factor if its join with any central factor is a central factor. Then, any central subgroup, cocentral subgroup, or direct factor is a join-transitively central factor. In other words, a join of two central factors of which one is a direct factor, a central subgroup, or a cocentral subgroup, is again a central factor. For full proof, refer: Direct factor implies join-transitively central factor, Central implies join-transitively central factor

On the other hand, if we define a join-transitively transitively normal subgroup as a subgroup whose join with any transitively normal subgroup is transitively normal, neither direct factors nor central subgroups are necessarily join-transitively transitively normal. Cocentral subgroups are join-transitively transitively normal.

Centralizer-closed

A centralizer-closed subgroup property is a subgroup property such that the centralizer of any subgroup satisfying the property also satisfies the property. Some of the subgroup properties here are centralizer-closed, and others are not.

Quotient-transitivity

A quotient-transitive subgroup property is a subgroup property $p$ such that whenever $H \le K \le G$ are such that $H$ is normal in $G$, $H$ satisfies $p$ in $G$ and $K/H$ satisfies $p$ in $G/H$, then $K$ satisfies $p$ in $G$. Some of the subgroup properties here are quotient-transitive, and others are not: