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

By induction, we have:

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