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

## Contents

## Definition

The **in-normalizer operator** is a map from the subgroup property space to itself that takes as input a subgroup property and outputs a subgroup property defined as follows: satisfies property in if and only if satisfies property in .

## 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 are subgroup properties and is the operator, then *

If the the in-normalizer of is also the in-normalizer of .

### 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 is stronger than the property of being a normal subgroup, then is stronger than the in-normalizer of . In any case, the conjunction of with the property of being normal, is stronger than the in-normalizer of .

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.