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

• A cyclic group generated by g is

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

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

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

## A cyclic group of order n is Isomorphic to the integers modulo n with addition

Abstract Algebra/Group Theory/Group/A cyclic group of order n is Isomorphic to the integers modulo n with addition