# NSCFN-subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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]

## Definition

### Symbol-free definition

A subgroup of a group is termed a NSCFN-subgroup if it is normal, fully normalized and self-centralizing in the whole group.

### Definition with symbols

A subgroup $H$ of a group $G$ is termed a NSCFN-subgroup if $H \triangleleft G$, $H$ is fully normalized in $G$, and $C_G(H) \le H$.

Equivalently, $H$ is a NSCFN-subgroup of $G$ if $H$ is a normal subgroup of $G$, and the induced homomorphism $G \to \operatorname{Aut}(H)$ induced by the conjugation action is surjective with kernel equal to $Z(H)$.