The subgroup $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$, and cyclic groups

Let $G$ be a group (Group and abelian group) and $S \subseteq G$ a subset. The set of subgroups of $G$ containing $S$ is nonempty, since $G$ itself is such a subgroup, so its intersection is a subgroup of $G$ by The intersection of a nonempty family of subgroups of $G$ is a subgroup of $G$. That intersection is the subgroup generated by $S$,

$\langle S \rangle \;:=\; \bigcap \{\, H \;:\; H \le G \text{ and } S \subseteq H \,\} .$

It contains $S$, being an intersection of sets each containing $S$, and it is contained in every subgroup of $G$ that contains $S$; so it is the smallest subgroup of $G$ containing $S$, and these two properties determine it uniquely. The elements of $S$ are called generators.

For a single element $g \in G$ we write $\langle g \rangle := \langle \{g\}\rangle$ and call it the cyclic subgroup generated by $g$. A group $G$ is cyclic when $G = \langle g \rangle$ for some $g \in G$.

By convention $\langle \varnothing \rangle = \{e\}$: the trivial subgroup is the smallest subgroup containing the empty set, and this is a consequence of the definition, not a stipulation, since every subgroup contains $e$ (Subgroup).

• Two descriptions, one object. The definition above is "from outside": cut down from all subgroups containing $S$. There is also a description "from inside", as the set of all finite products of generators and their inverses. For a single generator that inside description is $\langle g \rangle = \{\, g^{n} : n \in \mathbb{Z} \,\}$, proved in $\langle g \rangle = \{\, g^{n} : n \in \mathbb{Z} \,\}$, and every cyclic group is abelian. The general case belongs to a later page; nothing here needs it.

• Cyclic does not mean finite. $(\mathbb{Z},+)$ is cyclic, generated by $1$, and infinite; the generator may also fail to be unique, since $-1$ generates it too.

• Every cyclic group is abelian ($\langle g \rangle = \{\, g^{n} : n \in \mathbb{Z} \,\}$, and every cyclic group is abelian), so a non-abelian group is never cyclic; the converse fails, and the Klein four-group on the companion page is an abelian group that is not cyclic.