Definition:Cyclic Group/Notation
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Definition
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.