Abstract Algebra/Group Theory/Subgroup/Cyclic Subgroup/Definition of a Cyclic Subgroup

Let G be a Group. Let g be an element of G.

The Cyclic Subgroup generated by g is:


 * $$ \forall \; g \in G: \langle g \rangle = \lbrace g ^{n} \; | \; n \in \mathbb{Z} \rbrace $$


 * where $$ g^{n} =

\begin{cases} \underbrace{g \ast g \cdots \ast g}_{n}, & n \in \mathbb{Z}, n \ge 0\\ \underbrace{g^{-1} \ast g^{-1} \cdots \ast g^{-1}}_{-n}, & n \in \mathbb{Z}, n < 0 \end{cases}

$$


 * By induction, we have:


 * $$ \forall g \in G: \forall n, m \in \mathbb{Z}: g^{m + n} = g^{m}  \ast g^{n}

\text{ and } g^{mn} = [g^{m}]^{n} $$