# Normality is strongly UL-intersection-closed

From Groupprops

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 is a group, is an indexing set, and for each , we have subgroups such that is normal in . Then, the intersection of the s is normal in the intersection of the s.

## 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 -subnormal, for any fixed , is also strongly intersection-closed.

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