# Proving intersection-closedness

This is a survey article related to:subgroup metaproperty satisfaction

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A subgroup property is termed an intersection-closed subgroup property if an arbitrary (nonempty) intersection of subgroups having property also has property . 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.

is termed a finite-intersection-closed subgroup property if the intersection of finitely many subgroups satisfying the property also has the property . 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:

- Disproving intersection-closedness
- All intersection-closed subgroup properties
- All finite-intersection-closed subgroup properties

## Contents

## Invariance properties

`Further information: Invariance implies strongly intersection-closed`

Suppose is a property of functions from a group to itself. The invariance property corresponding to is defined as the following property : has property in if every function from to itself satisfying property sends 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 is termed a left-hereditary subgroup property if, whenever is a subgroup of a group satisfying property in , any subgroup of also satisfies property in .

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

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

## Effect of logical operators

### Conjunction

`Further information: Intersection-closedness is conjunction-closed`

If and are both intersection-closed subgroup properties, so is the conjunction (**AND**) of and . 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 and , and outputs the property , defined as follows. A subgroup of a group satisfies property in if there are subgroups <math<K,L</math> of such that satisfies , satisfies in , and .

Here are some facts:

- Intersection operator preserves finite-intersection-closedness: If are both finite-intersection-closed, so is .
- Intersection operator preserves intersection-closedness: If are both intersection-closed, so is .

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 and be subgroup properties. The left residual of by is the following subgroup property : a subgroup of a group satisfies property in if, for any group containing as a subgroup with property , contains with property .

It turns out that if is intersection-closed, so is the left residual of by . 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 is defined as follows. For a subgroup property , is defined as follows: a subgroup of a group satisfies property if, for any subgroup of , satisfies property in .