# Subgroup generated by a subset

## Contents

## Definition

### Symbol-free definition

Given a group, and a subset of the group, the subgroup generated by that subset is defined in the following equivalent ways:

- It is the intersection of all subgroups containing that subset
- It is a subgroup containing that subset, such that the subset is a generating set for the subgroup

### Definition with symbols

Given a group and a subset of , the *subgroup generated* by is defined in the following equivalent ways:

- It is the intersection of all subgroups of containing
- It is a subgroup such that and is a generating set for

(Recall the fact that an intersection of subgroups is always a subgroup).

The subgroup generated by a subset is denoted .

## Examples

### Extreme examples

- If a subset of a group is a subgroup, then equals the subgroup generated by .
- The subgroup generated by the empty subset is the trivial subgroup: it comprises only the identity element.

### Other generic examples

- The subgroup generated by a single element is the set of all its powers. In other words, the subgroup generated by an element is the set of all elements expressible as . This is also termed the
*cyclic subgroup*generated by .

### Examples in Abelian groups

- In the group of integers under addition, the subgroup generated by the integers and is the subgroup of even integers.
- In the group of rational numbers under addition, the subgroup generated by the rational number is the group of integers and half-integers under addition.