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


 * A generated by g is


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


 * 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} $$


 * Induction shows: $$g^{m + n} = g^{m} \ast g^{n} \text{ and } g^{mn} = [g^{m}]^{n}$$