Intersection-closed subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article is about a general term. A list of important particular cases (instances) is available at Category: Intersection-closed subgroup properties

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?

Definition

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 $G$.

Relation with other metaproperties

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. Further information: strongly intersection-closed subgroup property

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. Further information: strongly finite-intersection-closed subgroup property

Stronger metaproperties

Metaproperties stronger than the metaproperty of being intersection-closed:

Metametaproperties

Conjunction-closedness

This subgroup metaproperty is conjunction-closed: an arbitrary conjunction (AND) of subgroup properties satisfying this metaproperty, also satisfies this metaproperty
View conjunction-closed subgroup metaproperties

Effect of left residuals

This subgroup metaproperty is left residual-preserved: the left residual of any subgroup property satisfying this metaproperty, by any subgroup property, also satisfies this metaproperty.
View a complete list of such metaproperties

Naturally arising intersection-closed subgroup properties

Because of invariance

Further information: Invariance property

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.

Properties that are not intersection-closed

Further information: Disproving intersection-closedness

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.