Cyclic subgroup generated by an element

Definition

Let $G$ be a group and let $g \in G$. The cyclic subgroup generated by $g$ is the set $\langle g \rangle = \{g^k : k \in \mathbb{Z}\}$ of all integer powers of $g$.

Remarks

By the laws of exponents, $g^j (g^k)^{-1} = g^{j-k}$, thus $\langle g \rangle$ is a subgroup of $G$. It is the subgroup generated by the set $\{g\}$. In additive notation, $\langle g \rangle = \{kg : k \in \mathbb{Z}\}$, for example $\langle 3 \rangle = 3\mathbb{Z}$ in $(\mathbb{Z}, +)$.

Explanations

L1 human

Take one element and all its powers, positive and negative, and the identity. These powers form a subgroup, the smallest one that contains the element. In the integers with addition, the cyclic subgroup generated by $3$ is the set of multiples of $3$.