Disproving intersection-closedness

This survey article is about proof techniques for or related to: subgroup metaproperty satisfaction
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This article lists some common techniques, ideas, and underlying themes, behind proofs that certain subgroup properties are not intersection-closed. While some of these techniques involve a direct construction ofcounterexamples, others simply use the fact that the subgroup does not satisfy various corollaries of intersection-closedness.

The direct product technique

Outline of the technique

This method is used to try to construct an example of an intersection of direct factors which does not satisfy a given property $p$ . It thus could work for properties that are implied by the property of being a direct factor but are not implied by the property of normality. (If it is implied by normality, then any intersection of direct factors would have the property).

The idea is to take the direct product of $G$ with an Abelian group $A$ , such that we have in mind a surjective homomorphism $ρ : G \to A$ with kernel a subgroup $N$ of $G$ .

Now we consider the automorphism $σ : G \times A \to G \times A$ defined as $(g, a) \mapsto (g, ρ (g) a)$ . Note that this automorphism fixes $A$ pointwise.

Consider the subgroup $G \cap σ (G)$ . This is precisely the kernel $N$ of $ρ$ . Thus, it suffices to show that $N$ does not satisfy the property $p$ in $G \times A$ .

How this technique is applied

Suppose $p$ is a property such that:

Any direct factor satisfies property $p$
$p$ satisfies the intermediate subgroup condition
There is a group with a normal subgroup that does not satisfy $p$ , and such that the quotient by that normal subgroup is Abelian.

Then, the above technique constructs for us an intersection of direct factors which does not satisfy the property $p$ .

Another way of saying this is that if a subgroup property is satisfied by direct factors and is intersection-closed, then it must be satisfied by every Abelian quotient-subgroup.

The core technique

The basic result

If a subgroup property $p$ is intersection-closed, then the normal core of any subgroup with property $p$ also has property $p$ . So does the characteristic core.

Thus, in particular, and intersection-closed subgroup property is normal core-closed as well as characteristic core-closed.

Application to NCI-subgroup properties

A subgroup property is termed a NCI-subgroup property if the only normal subgroup that satisfies it is the whole group. Clearly, if a subgroup property is NCI, then the only way it can be normal core-closed is if it is satisfied by the whole group only.

Thus, to show that a subgroup property is not normal core-closed, it suffices to show that it is a NCT-subgroup property and is satisfed by a subgroup other than the whole group.

Examples of properties for which this approach works: