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:

$\mathrm{\forall }g\in G:⟨g⟩=\{g^n|n\in \mathbb{Z}\}$

where $g^n=\{\begin{array}{cc}\underset{n}{\underbrace{g\ast g\cdots \ast g}},& n\in \mathbb{Z},n\ge 0\\ \underset{-n}{\underbrace{g^{-1}\ast g^{-1}\cdots \ast g^{-1}}},& n\in \mathbb{Z},n<0\end{array}$

By induction, we have:

$\mathrm{\forall }g\in G:\mathrm{\forall }n,m\in \mathbb{Z}:g^{m+n}=g^m\ast g^n\text{ and }{g}^{mn}=[g^m{]}^n$