Intersection-closed subgroup property

History
The observations that an arbitrary intersection of subgroups is a subgroup, and that an arbitrary intersection of normal subgroups is a normal subgroup, date back to the beginning of group theory. Whenever new subgroup properties were encountered, two questions occurred naturally:


 * Does a finite intersection of subgroups (each with the property) also have the property?
 * Does an arbitrary intersection of subgroups (each with the property) also have the property?

Symbol-free definition
A subgroup property $$p$$ is termed intersection-closed if the intersection of a nonempty (but otherwise arbitrary, possibly infinite) collection of subgroups, each with property $$p$$, also has property $$p$$.

Definition with symbols
A subgroup property $$p$$ is termed intersection-closed if whenever $$I$$ is a nonempty indexing set, and $$H_i$$ are subgroups of $$G$$ with $$i$$ ranging over $$I$$, such that each $$H_i$$ satisfies property $$p$$ in $$G$$, then the intersection of all the $$H_i$$s also satisfies $$p$$ in $$G$$.

A slight change in definition
A subgroup property closed under finite intersections of subgroups is termed a finite-intersection-closed subgroup property.

We call a subgroup property strongly intersection-closed if it is both intersection-closed and identity-true. The property of being strongly intersection-closed is equivalent to the property of being closed under arbitrary intersections, including the possibility of the empty intersection.

Similarly we define being strongly finite-intersection-closed as being both finite-intersection-closed and identity-true. This is the same as being t.i. with respect to the intersection operator.

Stronger metaproperties
Metaproperties stronger than the metaproperty of being intersection-closed:


 * Weaker than::Strongly intersection-closed subgroup property: See the above discussion
 * Weaker than::Invariance property: In fact, any invariance property is strongly intersection-closed
 * Weaker than::UL-intersection-closed subgroup property
 * Weaker than::Left-hereditary subgroup property: Any left-hereditary subgroup property is intersection-closed, though, unless it is the tautology, it is not strongly intersection-closed.

Weaker metaproperties

 * Stronger than::Finite-intersection-closed subgroup property
 * Stronger than::Conjugate-intersection-closed subgroup property
 * Stronger than::Normal core-closed subgroup property
 * Stronger than::Characteristic core-closed subgroup property

Related metaproperties

 * Join-closed subgroup property

Because of invariance
Any invariance property with respect to the function restriction formalism is invariance-closed. Thus, the properties of being normal, characteristic, fully characteristic, and strictly characteristic are all intersection-closed.

Because the subgroup property is left hereditary
Any left hereditary property is automatically intersection-closed.

Left antihereditary properties
The only way a left antihereditary property can be intersection-closed is if for every group, there is only one subgroup having that property. Since any minimal property and any maximal property is left antihereditary, this gives plenty of examples of properties that are not intersection-closed.

Properties opposing normality
We know that any intersection-closed property must be a normal core-closed subgroup property, and in fact, a characteristic core-closed subgroup property. Thus, to prove that a subgroup property is not intersection-closed, it suffices to find a subgroup satisfying the property, whose normal core, or characteristic core, does not satisfy the property.

In particular, any NCT-subgroup property or CCT-subgroup property that is satisfied by at least one proper subgroup is not normal core-closed (respectively characteristic core-closed) and hence it is not intersection-closed.

Direct factors and retracts
An intersection of direct factors need not be a direct factor. This is demonstrated by looking at the direct product of a cyclic group of order $$p$$ with a cyclic group of order $$p^2$$ and considering the intersection of any two automorphic copies of the cyclic group of order $$p^2$$.

Incidentally, this also shows that an intersection of two retracts may not be a retract.