Proving intersection-closedness

A subgroup property $$p$$ is termed an survey article about::intersection-closed subgroup property if an arbitrary (nonempty) intersection of subgroups having property $$p$$ also has property $$p$$. $$p$$ is termed a survey article about::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 survey article about::finite-intersection-closed subgroup property if the intersection of finitely many subgroups satisfying the property $$p$$ also has the property $$p$$. $$p$$ is a survey article about::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:


 * Disproving intersection-closedness

Invariance properties
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.
 * Characteristic subgroup: This is the invariance property with respect to automorphisms.
 * Strictly characteristic subgroup: This is the invariance property with respect to surjective endomorphisms.
 * Fully invariant subgroup: This is the invariance property with respect to endomorphisms.
 * Injective endomorphism-invariant subgroup: This is the invariance property with respect to injective endomorphisms.
 * Cofactorial automorphism-invariant subgroup: This is the invariance property with respect to cofactorial automorphisms.

Left-hereditary subgroup properties
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:


 * Central subgroup
 * Abelian subnormal subgroup
 * Nilpotent subnormal subgroup

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.

Conjunction
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 of $$G$$ such that $$K$$ satisfies p in $$G$$, $$L$$ 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
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
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 $$K$$.