Characteristicity is strongly intersection-closed

Verbal statement
An arbitrary (possibly empty) intersection of characteristic subgroups of a group is a characteristic subgroup.

Statement with symbols
Suppose $$I$$ is an indexing set and $$H_i, i \in I$$ is a collection of characteristic subgroups of a group $$G$$. Then, the intersection of subgroups $$\bigcap_{i \in I} H_i$$ is also a characteristic subgroup of $$G$$.

Generalizations

 * Invariance implies strongly intersection-closed: Any invariance property (i.e., a property that can be expressed as invariance under a certain collection of functions) is strongly intersection-closed: an arbitrary intersection of subgroups with the property again has the property.

Other particular cases of this general result are:


 * Normality is strongly intersection-closed
 * Strict characteristicity is strongly intersection-closed
 * Full characteristicity is strongly intersection-closed

Analogues in other algebraic structures

 * Characteristicity is strongly intersection-closed in Lie rings
 * Derivation-invariance is strongly intersection-closed

Related metaproperty satisfactions and dissatisfactions for characteristicity

 * Characteristicity is transitive: A characteristic subgroup of a characteristic subgroup is characteristic.
 * Characteristicity is strongly join-closed: The subgroup generated by a collection of characteristic subgroups is characteristic.
 * Characteristicity is not finite-relative-intersection-closed
 * Characteristicity does not satisfy intermediate subgroup condition
 * Characteristicity does not satisfy transfer condition