Commutator-in-center is intersection-closed

From Groupprops
Jump to: navigation, search
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
|
Property "Page" (as page type) with input value "{{{property}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
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.

Related facts