# Free-quotient 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

A normal subgroup of a group is termed a **free-quotient subgroup** if it satisfies the following equivalent conditions:

- The quotient group is a free group.
- It is a complemented normal subgroup and has a complement that is a free group.

## Relation with other properties

### Weaker properties

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

Complemented normal subgroup | ||||

Normal subgroup | |FULL LIST, MORE INFO |