# Cocentral subgroup of normalizer

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This is a variation of cocentral subgroup|Find other variations of cocentral subgroup |

## Definition

### Symbol-free definition

A subgroup of a group is said to be a cocentral subgroup of normalizer if it is a cocentral subgroup of its normalizer.

### Definition with symbols

A subgroup $H$ of a group $G$ is said to be a cocentral subgroup of normalizer if $HZ(N_G(H)) = N_G(H)$ where $Z(N_G(H))$ denotes the center of $N_G(H)$ and $N_G(H)$ denotes the normalizer of $H$.

## Formalisms

### In terms of the in-normalizer operator

This property is obtained by applying the in-normalizer operator to the property: cocentral subgroup
View other properties obtained by applying the in-normalizer operator

## Metaproperties

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties