Normalizing join-closedness is left residual-preserved

Property-theoretic statement
The fact about::left residual operator for composition of a normalizing join-closed subgroup property by any subgroup property is again intersection-closed.

Statement with symbols
Suppose $$p$$ is a normalizing join-closed subgroup property. In other words, if $$H, K \le G$$ are subgroups such that $$K$$ normalizes $$H$$ and both $$H$$ and $$K$$ satisfy $$p$$, then the join of subgroups $$HK$$ also satisfies $$p$$.

Suppose $$q$$ is any subgroup property. Let $$r$$ be the left residual of $$p$$ by $$q$$. Then, $$r$$ is also normalizing join-closed: if $$H, K \le G$$ are subgroups such that $$K$$ normalizes $$H$$, and both $$H$$ and $$K$$ satisfy $$r$$, the join of subgroups $$HK$$ also satisfies the property $$r$$.

Similar facts about left residual-preserved

 * Conjugate-join-closedness is left residual-preserved
 * Join-closedness is left residual-preserved
 * Finite-join-closedness is left residual-preserved
 * Finite-conjugate-join-closedness is left residual-preserved
 * Intersection-closedness is left residual-preserved
 * Finite-intersection-closedness is left residual-preserved
 * Permuting join-closedness is left residual-preserved

Related facts about right residual-preserved
Some related examples of right residual-preserved subgroup metaproperties:


 * Upper join-closedness is right residual-preserved
 * Intermediate subgroup condition is right residual-preserved