# Cyclic Groups

Another type of group is called the cyclic group, which is a group generated by a single element. For example $\mathbb {Z}_8$ is a cyclic group because all of the elements in $\mathbb{Z}_8$ can be generated by $1$ by modular arithmetic. $\mathbb{Z}_8 = <1> = \{1, 1+1, 1+1+1 ... \}$.

More formally let $G$ be a group and $g \in G$, $G$ is a cyclic group if $G = = \{g^n | n \in N\}$; $g^n$ does not mean exponentiation in that $g^2$ does not mean $g$ multiplied by $g$, it means $g*g$ where $*$ can be addition or any defined binary operation.  For example $g^5 = g*g*g*g*g$.

Another example is the group $H = \{1,-1,i,-i\}$ $= $ $= \{i, i^2, i^3, i^4\}$ $= \{i, -1, -i, 1\}$.

Filed Under: Crashes CoursesMathematics

Sources Used

• Abstract Algebra, Theory and Application by Thomas W. Judson
• Abstract Algebra by David S. Dummit and Richard M. Foote