In-normalizer operator

Definition
The in-normalizer operator is a map from the subgroup property space to itself that takes as input a subgroup property $$p$$ and outputs a subgroup property $$q$$ defined as follows: $$H$$ satisfies property $$q$$ in $$G$$ if and only if $$H$$ satisfies property $$p$$ in $$N_G(H)$$.

Application
Important instances of application of the in-normalizer operator:



Properties
If $$p \le q$$ the the in-normalizer of $$p$$ is also $$\le$$ the in-normalizer of $$q$$.

The in-normalizer operator is idempotent, in the sense that applying it twice to a given subgroup property has the same effect as applying it once. An element is a fixed-point under this operator if and only if it is a in-normalizer subgroup property.

Conditionally ascendant
If $$p$$ is stronger than the property of being a normal subgroup, then $$p$$ is stronger than the in-normalizer of $$p$$. In any case, the conjunction of $$p$$ with the property of being normal, is stronger than the in-normalizer of $$p$$.

This is true of all the examples mentioned above.

Identity-true implies self-normalizing
The property of being the improper subgroup (that is, being the whole group) gets mapped under the in-normalizer operator to the property of being a self-normalizing subgroup. This, along with the monotonicity, tells us that any identity-true subgroup rpoperty gets mapped under the in-normalizer operator to a property weaker than the property of being a self-normalizing subgroup.