Proving join-closedness

From Groupprops
Jump to: navigation, search
This is a survey article related to:subgroup metaproperty satisfaction
View other survey articles about subgroup metaproperty satisfaction

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:

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:

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:

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-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.