Normalizing join-closedness is left residual-preserved

This article gives the statement, and possibly proof, of a subgroup metaproperty (i.e., Normalizing join-closed subgroup property (?)) satisfying a subgroup metametaproperty (i.e., Left residual-preserved subgroup metaproperty (?))
View all subgroup metametaproperty satisfactions $|$ View all subgroup metametaproperty dissatisfactions

Statement

Property-theoretic statement

The 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 $H,K\leq G$ 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 $H,K\leq G$ are subgroups such that $K$ normalizes $H$, and both $H$ and $K$ satisfy $r$, the join of subgroups $HK$ also satisfies the property $r$.

Related facts

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