Proving join-closedness
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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.