Subgroup generated by a subset
- 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 .
- 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.