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A subgroup property is termed an intersection-closed subgroup property if an arbitrary (nonempty) intersection of subgroups having property also has property . is termed a strongly intersection-closed subgroup property if it is intersection-closed and is also an identity-true subgroup property -- it is satisfied by every group as a subgroup of itself.
is termed a finite-intersection-closed subgroup property if the intersection of finitely many subgroups satisfying the property also has the property . is a strongly finite-intersection-closed subgroup property if it is finite-intersection-closed and identity-true.
This article discusses techniques to prove that a given subgroup property is intersection-closed.
- Disproving intersection-closedness
- All intersection-closed subgroup properties
- All finite-intersection-closed subgroup properties
- 1 Invariance properties
- 2 Left-hereditary subgroup properties
- 3 Galois correspondences
- 4 Property of a normal subgroup based on the isomorphism class of its quotient group
- 5 Effect of logical operators
- 6 Effect of subgroup property modifiers
Further information: Invariance implies strongly intersection-closed
Suppose is a property of functions from a group to itself. The invariance property corresponding to is defined as the following property : has property in if every function from to itself satisfying property sends to within itself.
Invariance properties are strongly intersection-closed. In other words, they are closed under arbitrary intersections, and every group satisfies the property as a subgroup of itself.
Here are some examples:
- Normal subgroup: This is the invariance property with respect to inner automorphisms. Further information: Normality is strongly intersection-closed
- Characteristic subgroup: This is the invariance property with respect to automorphisms. Further information: Characteristicity is strongly intersection-closed
- Strictly characteristic subgroup: This is the invariance property with respect to surjective endomorphisms. Further information: Strict characteristicity is strongly intersection-closed
- Fully invariant subgroup: This is the invariance property with respect to endomorphisms. Further information: Full invariance is strongly intersection-closed
- Injective endomorphism-invariant subgroup: This is the invariance property with respect to injective endomorphisms. Further information: Injective endomorphism-invariance is strongly intersection-closed
- Cofactorial automorphism-invariant subgroup: This is the invariance property with respect to cofactorial automorphisms.
Left-hereditary subgroup properties
Further information: Left-hereditary implies intersection-closed
Left-hereditary subgroup properties are intersection-closed for obvious reasons. However, a left-hereditary subgroup property is not identity-true unless it is the tautology. Hence, it is not a strongly intersection-closed subgroup property.
Some examples are:
Some subgroup properties arise as a result of Galois correspondences. We cal such a property a Galois correspondence-closed subgroup property.
We start with a rule which, for every group, gives a binary relation between the group and another set constructed canonically from the group. The rule must be isomorphism-invariant, in the sense that any isomorphism of groups respects the binary relation.
The subgroup property we now get is the property of being a subgroup, which is also a closed subset of the group under the Galois correspondence induced by the binary relation.
Any Galois correspondence-closed subgroup property is strongly intersection-closed. Some examples are:
- C-closed subgroup: This corresponds to the relation between a group and itself by the commutativity relation. A c-closed subgroup is thus a subgroup that equals its double centralizer.
- Fixed-point subgroup of a subgroup of the automorphism group
Property of a normal subgroup based on the isomorphism class of its quotient group
- If is closed under taking finite subdirect products, then is finite-intersection-closed. In particular, for instance, if is a [quasivarietal group property]], is strongly finite-intersection-closed.
- If is closed under arbitrary subdirect products, then is intersection-closed. In particular, for instance, if is a varietal group property, is a strongly intersection-closed subgroup property.
Effect of logical operators
Further information: Intersection-closedness is conjunction-closed
If and are both intersection-closed subgroup properties, so is the conjunction (AND) of and . More generally, the conjunction of an arbitrary collection of intersection-closed subgroup properties is intersection-closed.
Note that in some cases, one of the properties in the conjunction is a group property interpreted as a subgroup property. In this case, it suffices to show that the group property is a subgroup-closed group property and the subgroup property is intersection-closed.
Analogous observations apply to strongly intersection-closed, finite-intersection-closed, and strongly finite-intersection-closed subgroup properties.
Some examples of conjunctions of intersection-closed properties that continue to be intersection-closed:
Effect of subgroup property modifiers
In this section, we discuss various subgroup property modifiers and their impact on the metaproperty of being intersection-closed and finite-intersection-closed.
By definition, the intersection-transiter of any subgroup property is a strongly finite-intersection-closed subgroup property. In other words, it is satisfied by every group as a subgroup of itself and is closed under all finite intersections.
Here are some facts:
- The finite-intersection-closure operator outputs a finite-intersection-closed subgroup property.
- The intersection-closure operator outputs an intersection-closed subgroup property.
The intersection operator takes as input two subgroup properties and , and outputs the property , defined as follows. A subgroup of a group satisfies property in if there are subgroups <math<K,L</math> of such that satisfies , satisfies in , and .
Here are some facts:
- Intersection operator preserves finite-intersection-closedness: If are both finite-intersection-closed, so is .
- Intersection operator preserves intersection-closedness: If are both intersection-closed, so is .
Analogous results hold for strongly finite-intersection-closed and strongly intersection-closed.
Left residual, left transiter
Let and be subgroup properties. The left residual of by is the following subgroup property : a subgroup of a group satisfies property in if, for any group containing as a subgroup with property , contains with property .
It turns out that if is intersection-closed, so is the left residual of by . Analogous observations hold for strongly intersection-closed, finite-intersection-closed.
The left transiter of a subgroup property is its left residual by itself. The above result shows that the left transiter of any intersection-closed subgroup property is intersection-closed.
Here are some examples:
- Normality is strongly intersection-closed, left transiter of normal is characteristic, characteristicity is strongly intersection-closed
- 2-subnormality is strongly intersection-closed, and hence, the property of being a left-transitively 2-subnormal subgroup is also strongly intersection-closed.
Transfer condition operator
Further information: Transfer condition operator preserves intersection-closedness