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

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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} = 

  \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

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}