Subgroup generated by a subset

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 $G$ and a subset $S$ of $G$, the subgroup generated by $S$ is defined in the following equivalent ways:

• It is the intersection of all subgroups of $G$ containing $S$
• It is a subgroup $H\le G$ such that $S\subset H$ and $S$ is a generating set for $H$

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

The subgroup generated by a subset $S$ is denoted $⟨S⟩$.

Examples

Extreme examples

• If a subset $S$ of a group $G$ is a subgroup, then $S$ equals the subgroup generated by $S$.
• 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 $g$ is the set of all elements expressible as $g^n,n\in \mathbb{Z}$. This is also termed the cyclic subgroup generated by $g$.

Examples in Abelian groups

• In the group of integers under addition, the subgroup generated by the integers $4$ and $6$ is the subgroup of even integers.
• In the group of rational numbers under addition, the subgroup generated by the rational number $1/2$ is the group of integers and half-integers under addition.