# Proving join-closedness

This is a survey article related to:subgroup metaproperty satisfaction

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A subgroup property is termed a join-closed subgroup property if the join of subgroups of a group, each of which satisfies property in the group, also satisfies in the group.

is a strongly join-closed subgroup property if it is both a join-closed subgroup property and a trivially true subgroup property: it is satisfied by the trivial subgroup in any group.

is a finite-join-closed subgroup property if it is closed under joins of finitely many subgroups. It is a strongly finite-join-closed subgroup property if it is both finite-join-closed and trivially true.

This article discusses methods to prove that a subgroup property is join-closed.

Also refer:

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

## Contents

## Endo-invariance properties

`Further information: Endo-invariance implies strongly join-closed`

Suppose is a property of functions from a group to itself. The invariance property corresponding to is the subgroup property defined as follows: satisfies in if, for any function from to itself satisfying property , .

An endo-invariance property is an invariance property corresponding to a property of functions that is only true for endomorphisms.

It turns out that any endo-invariance property is strongly join-closed: it is satisfied by the trivial subgroup and an arbitrary join of subgroups satisfying the property also satisfies the property.

Here are some examples:

- Normality is strongly join-closed: The property of being a normal subgroup is the endo-invariance property with respect to the property of being an inner automorphism.
- Characteristicity is strongly join-closed: The property of being a characteristic subgroup is the endo-invariance property with respect to the property of being an automorphism.
- Strict characteristicity is strongly join-closed: The property of being a strictly characteristic subgroup is the endo-invariance property with respect to the property of being a surjective endomorphism.
- Full invariance is strongly join-closed: The property of being a fully invariant subgroup is the endo-invariance property with respect to the property of being an endomorphism.

## Upward-closed subgroup properties

A subgroup property is termed an upward-closed subgroup property if, whenever is a subgroup of satisfying the property, any subgroup of containing also satisfies the property. Here are some examples:

## Effect of logical operators

### Conjunction

The conjunction (**AND**) of finite-join-closed subgroup properties is again finite-join-closed. Similarly, the conjunction of
join-closed subgroup properties is also join-closed.

### Disjunction

The disjunction (**OR**) of join-closed subgroup properties need not be join-closed. Similarly, the disjunction of finite-join-closed subgroup properties need not be finite-join-closed.

## Effect of subgroup property modifiers

### Join-transiter

Suppose is a subgroup property. The join-transiter of is the property of being a subgroup whose join with any subgroup having property still has property .

The join-transiter of any subgroup property is a strongly finite-join-closed subgroup property.

Some examples are:

- Join-transitively subnormal subgroup: This is the join-transiter of the property of being a subnormal subgroup.
- Join-transitively finite subgroup: This is the join-transiter of the property of being a finite subgroup.
- Join-transitively pronormal subgroup: This is the join-transiter of the property of being a pronormal subgroup.

### Join-closure

The finite-join-closure operator takes as input a subgroup property and outputs the property of being a join of finitely many subgroups satisfying that property. The finite-join-closure operator applied to any property gives a finite-join-closed subgroup property.

The join-closure operator takes as input a subgroup property and outputs the property of being a join of (possibly infinitely) many subgroups satisfying the property. The join-closure operator applied to any property gives a join-closed subgroup property.

### Join operator

The join operator is a binary operator that takes as input two subgroup properties. Suppose are subgroup properties. A subgroup of a group is said to satisfy the property obtained by the join operator on and if there exist subgroups such that .

The following are true:

- The join operator applied to two finite-join-closed subgroup properties is also finite-join-closed.
- The join operator applied to two join-closed subgroup properties is also join-closed.