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Cyclic Groups

Basic definitions and examples

See these notes and video.

  • Let $G$ be a group and $g\in G$. Then \(< g > = \{g^{k} : k\in\mathbb{Z}\}\) is a subgroup of $G$ called the cyclic subgroup of $G$ generated by $g$.

  • A group $G$ is cyclic if there is $g\in G$ such that $G= < g >$. In this case, we say that $g$ generates $G$.

  • Examples of cyclic groups: $\mathbb{Z}$, $\mathbb{Z}_{n}$, $U(7)$, rotations of an equilateral triangle.

  • Examples of noncyclic groups: Symmetries of an equilateral triangle, quaternion group.

Properties of cyclic subgroups and groups

See these notes and video.

  • The subset generated by an element of a group is a subgroup.

  • Cyclic groups are abelian.

  • Every subgroup of a cyclic group is cyclic.

Orders of elements

Note: Please review Euclid’s Algorithm. See video and notes for the theory and this video for a numerical example.

You may also find it helpful to look at how one uses Euclid’s algorithm to solve congruence equations.

See video and notes for discussion of this material.

  • The order of an element $g\in G$ is the number of elements in $\langle g \rangle$.

  • Suppose that $G$ is a cyclic group of order $n$, and that $a$ is a generator of $G$. Then $a^{k}=e$ if and only if $n$ divides $k$.

  • Let $G$ be a cyclic group of order $n$ and suppose that $a$ is a generator of $G$. If $b=a^{k}$, then the order of $b$ is $n/d$ where $d=\mathrm{gcd}(k,n)$.

  • A congruence class $[r]$ generates $\mathbb{Z}_{n}$ if and only if $\mathrm{gcd}(r,n)=1$. More generally, if $G$ is a cyclic group of order $n$ generated by $g$, then $g^r$ is a generator of $G$ if and only if $\mathrm{gcd}(r,n)=1$.

Further examples and results.

See notes and video.

  • For any $n>0$, the complex solutions to the polynomial $z^{n}=1$ form a cyclic group of order $n$ called the group of $n^{th}$ roots of unity. A generator of the group of $n^{th}$ roots of unity is called a primitive $n^{th}$ root of unity.