Proving intersection-closedness

This is a survey article related to:subgroup metaproperty satisfaction
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A subgroup property $p$ is termed an intersection-closed subgroup property if an arbitrary (nonempty) intersection of subgroups having property $p$ also has property $p$ . $p$ is termed a strongly intersection-closed subgroup property if it is intersection-closed and is also an identity-true subgroup property -- it is satisfied by every group as a subgroup of itself.

$p$ is termed a finite-intersection-closed subgroup property if the intersection of finitely many subgroups satisfying the property $p$ also has the property $p$ . $p$ is a strongly finite-intersection-closed subgroup property if it is finite-intersection-closed and identity-true.

This article discusses techniques to prove that a given subgroup property is intersection-closed.

Also refer:

Invariance properties

Further information: Invariance implies strongly intersection-closed

Suppose $a$ is a property of functions from a group to itself. The invariance property corresponding to $a$ is defined as the following property $p$ : $H$ has property $p$ in $G$ if every function from $G$ to itself satisfying property $a$ sends $H$ to within itself.

Invariance properties are strongly intersection-closed. In other words, they are closed under arbitrary intersections, and every group satisfies the property as a subgroup of itself.

Here are some examples:

Normal subgroup: This is the invariance property with respect to inner automorphisms. Further information: Normality is strongly intersection-closed
Characteristic subgroup: This is the invariance property with respect to automorphisms. Further information: Characteristicity is strongly intersection-closed
Strictly characteristic subgroup: This is the invariance property with respect to surjective endomorphisms. Further information: Strict characteristicity is strongly intersection-closed
Fully invariant subgroup: This is the invariance property with respect to endomorphisms. Further information: Full invariance is strongly intersection-closed
Injective endomorphism-invariant subgroup: This is the invariance property with respect to injective endomorphisms. Further information: Injective endomorphism-invariance is strongly intersection-closed
Cofactorial automorphism-invariant subgroup: This is the invariance property with respect to cofactorial automorphisms.

Left-hereditary subgroup properties

Further information: Left-hereditary implies intersection-closed

A subgroup property $p$ is termed a left-hereditary subgroup property if, whenever $H$ is a subgroup of a group $G$ satisfying property $p$ in $G$ , any subgroup $K$ of $H$ also satisfies property $p$ in $G$ .

Left-hereditary subgroup properties are intersection-closed for obvious reasons. However, a left-hereditary subgroup property is not identity-true unless it is the tautology. Hence, it is not a strongly intersection-closed subgroup property.

Some examples are:

Galois correspondences

Some subgroup properties arise as a result of Galois correspondences. We cal such a property a Galois correspondence-closed subgroup property.

We start with a rule which, for every group, gives a binary relation between the group and another set constructed canonically from the group. The rule must be isomorphism-invariant, in the sense that any isomorphism of groups respects the binary relation.

The subgroup property we now get is the property of being a subgroup, which is also a closed subset of the group under the Galois correspondence induced by the binary relation.

Any Galois correspondence-closed subgroup property is strongly intersection-closed. Some examples are:

C-closed subgroup: This corresponds to the relation between a group and itself by the commutativity relation. A c-closed subgroup is thus a subgroup that equals its double centralizer.
Fixed-point subgroup of a subgroup of the automorphism group

Property of a normal subgroup based on the isomorphism class of its quotient group

Suppose $a$ is a group property and $p$ is the property of being a normal subgroup of a group for which the quotient group has property $a$ . Then:

If $a$ is closed under taking finite subdirect products, then $p$ is finite-intersection-closed. In particular, for instance, if $a$ is a [quasivarietal group property]], $p$ is strongly finite-intersection-closed.
If $a$ is closed under arbitrary subdirect products, then $p$ is intersection-closed. In particular, for instance, if $a$ is a varietal group property, $p$ is a strongly intersection-closed subgroup property.

Effect of logical operators

Conjunction

Further information: Intersection-closedness is conjunction-closed

If $p$ and $q$ are both intersection-closed subgroup properties, so is the conjunction (AND) of $p$ and $q$ . More generally, the conjunction of an arbitrary collection of intersection-closed subgroup properties is intersection-closed.

Note that in some cases, one of the properties in the conjunction is a group property interpreted as a subgroup property. In this case, it suffices to show that the group property is a subgroup-closed group property and the subgroup property is intersection-closed.

Analogous observations apply to strongly intersection-closed, finite-intersection-closed, and strongly finite-intersection-closed subgroup properties.

Some examples of conjunctions of intersection-closed properties that continue to be intersection-closed:

c-closed normal subgroup is the conjunction of c-closed subgroup and normal subgroup.

Effect of subgroup property modifiers

In this section, we discuss various subgroup property modifiers and their impact on the metaproperty of being intersection-closed and finite-intersection-closed.

Intersection-transiter

By definition, the intersection-transiter of any subgroup property is a strongly finite-intersection-closed subgroup property. In other words, it is satisfied by every group as a subgroup of itself and is closed under all finite intersections.

Intersection-closure operator

Here are some facts:

The finite-intersection-closure operator outputs a finite-intersection-closed subgroup property.
The intersection-closure operator outputs an intersection-closed subgroup property.

Intersection operator

The intersection operator takes as input two subgroup properties $p$ and $q$ , and outputs the property $p\cap q$ , defined as follows. A subgroup $H$ of a group $G$ satisfies property $p\cap q$ in $G$ if there are subgroups <math<K,L</math> of $G$ such that $K$ satisfies $p<math>in<math>G$ , $L$ satisfies $q$ in $G$ , and $H=K\cap L$ .

Here are some facts:

Intersection operator preserves finite-intersection-closedness: If $p,q$ are both finite-intersection-closed, so is $p\cap q$ .
Intersection operator preserves intersection-closedness: If $p,q$ are both intersection-closed, so is $p\cap q$ .

Analogous results hold for strongly finite-intersection-closed and strongly intersection-closed.

Left residual, left transiter

Further information: Intersection-closedness is left residual-preserved, left transiter preserves intersection-closedness

Let $p$ and $q$ be subgroup properties. The left residual of $p$ by $q$ is the following subgroup property $r$ : a subgroup $H$ of a group $G$ satisfies property $r$ in $G$ if, for any group $K$ containing $G$ as a subgroup with property $p$ , $K$ contains $H$ with property $p$ .

It turns out that if $p$ is intersection-closed, so is the left residual of $p$ by $q$ . Analogous observations hold for strongly intersection-closed, finite-intersection-closed.

The left transiter of a subgroup property is its left residual by itself. The above result shows that the left transiter of any intersection-closed subgroup property is intersection-closed.

Here are some examples:

Normality is strongly intersection-closed, left transiter of normal is characteristic, characteristicity is strongly intersection-closed
2-subnormality is strongly intersection-closed, and hence, the property of being a left-transitively 2-subnormal subgroup is also strongly intersection-closed.

Transfer condition operator

Further information: Transfer condition operator preserves intersection-closedness

The transfer condition operator $T$ is defined as follows. For a subgroup property $p$ , $T(p)$ is defined as follows: a subgroup $H$ of a group $G$ satisfies property $T(p)$ if, for any subgroup $K$ of $G$ , $H\cap K$ satisfies property $T(p)$ in $K$ .