# Quotient-isomorph-containing subgroup

From Groupprops

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]

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

## Definition

Suppose is a group and is a subgroup. We say that is a quotient-isomorph-containing subgroup of if the following is true: is a normal subgroup of , and if is a normal subgroup of such that the quotient groups and are isomorphic groups, then .

If 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 | ||

quotient-homomorph-containing subgroup | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

characteristic subgroup | no other automorphic subgroup. | Template:Intermediate notion short |

### Related properties

- Isomorph-containing subgroup is a subgroup that contains every isomorphic subgroup to it in the whole group.