View source for 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 <span style="color:green;">Cyclic Subgroup </span> ''generated'' by ''g'' is:

: <math> \forall \; g \in G: \langle g \rangle = \lbrace g ^{n} \; | \;  n \in \mathbb{Z} \rbrace </math>

:: where <math> 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}

</math>

:: By induction, we have:

::: <math> \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}
</math>

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