Commutator-in-center is intersection-closed

This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Property "Page" (as page type) with input value "{{{metaproperty}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.

Statement

Property-theoretic statement

The property of being a commutator-in-center subgroup is an intersection-closed subgroup property: it is closed under arbitrary nonempty intersections. Note that it is not a strongly intersection-closed subgroup property because it is not closed under the empty intersection, since every group need not satisfy this property within itself.

Statement with symbols

Suppose $G$ is a group and $H_i, i \in I$ is a nonempty collection of commutator-in-center subgroups of $G$. Let $H = \bigcap_{i \in I} H_i$. Then, $H$ is also a commutator-in-center subgroup of $G$.