Definition:Cyclic Group

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Definition

A group $G$ is cyclic if there exists $g \in G$ such that for every $h \in G$, $h = g^n$ for some integer $n$.

That is, if every element of $G$ is a power of a fixed element of that group.

We say that $g$ generates $G$ and write $G = \left \langle {g}\right \rangle$.


It follows directly from List of Elements in Finite Cyclic Group that if $\left|{G}\right| = m$ then $G = \left\{{e, g, g^2, \ldots, g^{m - 1}}\right\}$ and $\left|{g}\right| = m$.


Notation

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

Some sources use the notation $\left[{g}\right]$ or $\left\langle{g}\right\rangle$ to denote the cyclic group generated by $g$.


As the Additive Group of Integers Modulo m is a cyclic group, the notation $\Z_m$ is often used.

This is justified as, from Cyclic Groups Same Order 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.


Group Presentation

The presentation of a finite cyclic group is:

$C_n = \left \langle {a: a^n = e} \right \rangle$


Also see

  • Results about cyclic groups can be found here.


Sources

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