In-normalizer operator

This article defines a subgroup property modifier (a unary subgroup property operator) -- viz an operator that takes as input a subgroup property and outputs a subgroup property

View a complete list of subgroup property modifiers OR View a list of all subgroup property operators (possibly with multiple inputs)

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:

• central factor of normalizer: obtained from central factor
• very weakly cocentral subgroup: obtained from cocentral subgroup
• AQIN-subgroup: obtained from Abelian-quotient subgroup

Properties

Monotonicity

This subgroup property modifier is monotone, viz if $p \le q$ are subgroup properties and $f$ is the operator, then $f(p) \le f(q)$

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

Idempotence

This subgroup property modifier is idempotent, viz applying it twice to a subgroup property has the same effect as applying it once

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.