# Quotient-isomorph-containing subgroup

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]

## Contents

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

## Definition

Suppose $G$ is a group and $H$ is a subgroup. We say that $H$ is a quotient-isomorph-containing subgroup of $G$ if the following is true: $H$ is a normal subgroup of $G$, and if $K$ is a normal subgroup of $G$ such that the quotient groups $G/H$ and $G/K$ are isomorphic groups, then $H \le K$.

If $G$ is a finite group, this property is equivalent to being a quotient-isomorph-free subgroup.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
quotient-isomorph-free subgroup normal and no other normal subgroup has an isomorphic quotient group. |FULL LIST, MORE INFO