Proving join-closedness

A subgroup property $$p$$ is termed a survey article about::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 survey article about::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 survey article about::finite-join-closed subgroup property if it is closed under joins of finitely many subgroups. It is a survey article about::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

Endo-invariance properties
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

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.

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.