Definition:Cyclic Group/Definition 1

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The group $G$ is cyclic if and only if every element of $G$ can be expressed as the power of one element of $G$:

$\exists g \in G: \forall h \in G: h = g^n$

for some $n \in \Z$.


A cyclic group with $n$ elements is often denoted $C_n$.

Some sources use the notation $\sqbrk g$ or $\gen g$ to denote the cyclic group generated by $g$.

From Integers Modulo m under Addition form Cyclic Group, $\struct {\Z_m, +_m}$ is a cyclic group.

Thus $\struct {\Z_m, +_m}$ often taken as the archetypal example of a cyclic group, and the notation $\Z_m$ is used.

This is justified as, from Cyclic Groups of Same Order are Isomorphic, $\Z_m$ is isomorphic to $C_m$.

In certain contexts $\Z_m$ is particularly useful, as it allows results about cyclic groups to be demonstrated using number theoretical techniques.

Also see

  • Results about cyclic groups can be found here.