# Intermediately normality-large 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 intermediately normality-large if it satisfies the following equivalent conditions:

1. It is normality-large in every intermediate subgroup
2. It does not occur as a retract of any subgroup properly containing it.

### Definition with symbols

A subgroup $H$ of a group $G$ is termed intermediately normality-large in $G$ if it satisfies the following equivalent conditions:

1. For any subgroup $K$ of $G$ containing $H$, and any normal subgroup $N$ of $K$ such that $N \cap H$ is trivial, $N$ must be trivial
2. If $K$ is a subgroup of $G$ containing $H$, and $\alpha$ is a retraction from $K$ to $H$ (i.e. a surjective homomorphism that restricts to the identity map on $H$), then $K = H$.