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:

• Direct factor: This is a subgroup that is normal and has a normal complement.
• Central subgroup: This is a subgroup contained in the center.
• Cocentral subgroup: This is a subgroup whose product with the center is the whole group.

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 properties central factor, conjugacy-closed normal subgroup, SCAB-subgroup, and transitively normal subgroup are all transitive. This is easily seem from the fact that they are balanced with respect to the function restriction formalism: they all have function restriction expressions with the left and right side equal. For full proof, refer: Balanced implies transitive
• Direct factor is transitive
• Cocentrality is transitive
• Any subgroup of a central subgroup is central, so centrality is transitive.
• Subset-conjugacy-closedness is transitive, conjugacy-closedness is transitive
• Retract is transitive

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:

• The properties central factor, conjugacy-closed normal subgroup, SCAB-subgroup, and transitively normal subgroup all satisfy the intermediate subgroup condition. This is because they all have function restriction expressions where the property on the left is inner automorphism. In particular, they are left-inner. For full proof, refer: Left-inner implies intermediate subgroup condition
• Direct factor satisfies intermediate subgroup condition
• Cocentrality satisfies intermediate subgroup condition
• Centrality satisfies intermediate subgroup condition
• Subset-conjugacy-closedness satisfies intermediate subgroup condition, Conjugacy-closedness satisfies intermediate subgroup condition
• Retract satisfies 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:

• Centrality satisfies transfer condition

Here are the many that do not satisfy transfer condition:

• Direct factor does not satisfy transfer condition, Cocentrality does not satisfy transfer condition
• Central factor does not satisfy transfer condition, transitive normality does not satisfy transfer condition, SCAB does not satisfy transfer condition (all rely on essentially the same example).
• Conjugacy-closedness does not satisfy transfer condition, Subset-conjugacy-closedness does not satisfy transfer condition, Retract does 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:

• Central factor satisfies image condition
• SCAB satisfies image condition
• Transitive normality satisfies image condition
• Direct factor satisfies image condition
• Centrality satisfies image condition
• Cocentrality satisfies image condition
• Conjugacy-closedness satisfies image condition
• Subset-conjugacy-closedness satisfies image condition
• Retract satisfies image condition

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.

• Direct factor is not finite-intersection-closed
• Central factor is not finite-intersection-closed, Transitive normality is not finite-intersection-closed, SCAB is not finite-intersection-closed, Conjugacy-closed normality is not finite-intersection-closed
• Conjugacy-closedness is not finite-intersection-closed, Subset-conjugacy-closedness is not finite-intersection-closed
• Cocentrality is not finite-intersection-closed

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):

• Central subgroup is closed under joins, because central subgroups are precisely subgroups contained in the center.
• Cocentral subgroup is closed under (nonempty) joins, because cocentrality is upward-closed: any subgroup containing a cocentral subgroup is cocentral.
• Direct factor is not finite-join-closed
• Central factor is not finite-join-closed, SCAB is not finite-join-closed, Conjugacy-closed normality is not finite-join-closed, Transitive normality is not finite-join-closed
• Conjugacy-closedness is not finite-join-closed, Retract is not finite-join-closed

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.

• Direct factor is not centralizer-closed
• Central factor is centralizer-closed
• Transitive normality is not centralizer-closed

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:

• Direct factor is quotient-transitive, Cocentrality is quotient-transitive
• Centrality is not quotient-transitive, Central factor is not quotient-transitive
• Transitive normality is not quotient-transitive
• Retract is quotient-transitive