Proving join-closedness

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

$p$ 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.

$p$ 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

Endo-invariance properties

Further information: Endo-invariance implies strongly join-closed

Suppose $a$ is a property of functions from a group to itself. The invariance property corresponding to $a$ is the subgroup property $p$ defined as follows: $H$ satisfies $p$ in $G$ if, for any function $f$ from $G$ to itself satisfying property $a$, $f(H) \subseteq H$.

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 $H$ is a subgroup of $G$ satisfying the property, any subgroup of $G$ containing $H$ also satisfies the property. Here are some examples:

• Cocentral subgroup
• Abelian-quotient subgroup

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

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 $p,q$ are subgroup properties. A subgroup $H$ of a group $G$ is said to satisfy the property obtained by the join operator on $p$ and $q$ if there exist subgroups $K,L \le G$ such that $H = \langle K, L \rangle$.

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.