#
nLab

free group

### Context

#### Group Theory

**group theory**

### Classical groups

### Finite groups

### Group schemes

### Topological groups

### Lie groups

### Super-Lie groups

### Higher groups

### Cohomology and Extensions

# Contents

## Idea

The **free group** on a given set $S$ is the free object on $S$ in the category of groups. The elements of $S$ are called the **generators** of this group.

## Properties

Every subgroup of a free group is itself a free group. This is the *Nielsen-Schreier theorem*.

Last revised on October 9, 2012 at 22:09:04.
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