Proving intersection-closedness

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This is a survey article related to:subgroup metaproperty satisfaction
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A subgroup property p is termed an intersection-closed subgroup property if an arbitrary (nonempty) intersection of subgroups having property p also has property p. p 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.

p is termed a finite-intersection-closed subgroup property if the intersection of finitely many subgroups satisfying the property p also has the property p. p 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.

Also refer:

Invariance properties

Further information: Invariance implies strongly intersection-closed

Suppose a is a property of functions from a group to itself. The invariance property corresponding to a is defined as the following property p: H has property p in G if every function from G to itself satisfying property a sends H 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:

Left-hereditary subgroup properties

Further information: Left-hereditary implies intersection-closed

A subgroup property p is termed a left-hereditary subgroup property if, whenever H is a subgroup of a group G satisfying property p in G, any subgroup K of H also satisfies property p in G.

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:

Galois correspondences

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:

Property of a normal subgroup based on the isomorphism class of its quotient group

Suppose a is a group property and p is the property of being a normal subgroup of a group for which the quotient group has property a. Then:

  • If a is closed under taking finite subdirect products, then p is finite-intersection-closed. In particular, for instance, if a is a [quasivarietal group property]], p is strongly finite-intersection-closed.
  • If a is closed under arbitrary subdirect products, then p is intersection-closed. In particular, for instance, if a is a varietal group property, p is a strongly intersection-closed subgroup property.

Effect of logical operators

Conjunction

Further information: Intersection-closedness is conjunction-closed

If p and q are both intersection-closed subgroup properties, so is the conjunction (AND) of p and q. 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.

Intersection-transiter

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.

Intersection-closure operator

Here are some facts:

Intersection operator

The intersection operator takes as input two subgroup properties p and q, and outputs the property p \cap q, defined as follows. A subgroup H of a group G satisfies property p \cap q in G if there are subgroups <math<K,L</math> of G such that K satisfies p<math> in <math>G, L satisfies q in G, and H = K \cap L.

Here are some facts:

Analogous results hold for strongly finite-intersection-closed and strongly intersection-closed.

Left residual, left transiter

Further information: Intersection-closedness is left residual-preserved, left transiter preserves intersection-closedness

Let p and q be subgroup properties. The left residual of p by q is the following subgroup property r: a subgroup H of a group G satisfies property r in G if, for any group K containing G as a subgroup with property p, K contains H with property p.

It turns out that if p is intersection-closed, so is the left residual of p by q. 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:

Transfer condition operator

Further information: Transfer condition operator preserves intersection-closedness

The transfer condition operator T is defined as follows. For a subgroup property p, T(p) is defined as follows: a subgroup H of a group G satisfies property T(p) if, for any subgroup K of G, H \cap K satisfies property T(p) in K.