# 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:

### The properties themselves

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

### 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: