Normality is strongly UL-intersection-closed

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This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) satisfying a subgroup metaproperty (i.e., strongly UL-intersection-closed subgroup property)
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Statement

Statement with symbols

Suppose G is a group, I is an indexing set, and for each i \in I, we have subgroups H_i \le K_i \le G such that H_i is normal in K_i. Then, the intersection of the H_is is normal in the intersection of the K_is.

Related facts

Applications

Combining this with the fact that UL-intersection-closedness is a [[composition-closed subgroup metaproperty, we can conclude that the property of being k-subnormal, for any fixed k, is also strongly intersection-closed.

Further information: UL-intersection-closedness is composition-closed

Weaker facts