Normality is strongly UL-intersection-closed

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 $H_{i}\leq K_{i}\leq G$ such that $H_{i}$ is normal in $K_{i}$. Then, the intersection of the $H_{i}$s is normal in the intersection of the $K_{i}$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 $k$-subnormal, for any fixed $k$, is also strongly intersection-closed.

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

Weaker facts

• Normality is strongly intersection-closed
• Normality satisfies transfer condition
• Normality satisfies intermediate subgroup condition