Group Theory Cyclic & Dihedral Groups Cheat Sheet

Key Facts

• A group $G$ is cyclic if there exists $a \in G$ such that $G = \langle a \rangle = \{a^k : k \in \mathbb{Z}\}$.
• If $|a| = n$, then $\langle a \rangle = \{e,a,a^2,\dots,a^{n-1}\}$ and $a^m = e$ exactly when $n \mid m$.
• In a cyclic group of order $n$, the order of $a^k$ is $|a^k| = \frac{n}{\gcd(n,k)}$.
• The element $a^k$ generates a cyclic group of order $n$ exactly when $\gcd(n,k) = 1$.
• Every subgroup of a cyclic group is cyclic, and for each divisor $d \mid n$ there is exactly one subgroup of order $d$ in $\mathbb{Z}_n$.
• The dihedral group of a regular $n$-gon has presentation $D_n = \langle r,s \mid r^n = e,\ s^2 = e,\ srs = r^{-1} \rangle$ and order $|D_n| = 2n$.
• Every element of $D_n$ has exactly one of the forms $r^k$ or $sr^k$, where $0 \le k < n$.
• In $D_n$, the main multiplication rules are $r^a r^b = r^{a+b}$, $r^a sr^b = sr^{b-a}$, and $sr^a sr^b = r^{b-a}$, with exponents taken modulo $n$.

Vocabulary

Cyclic group

A group generated by a single element, so every element can be written as $a^k$ for some $k \in \mathbb{Z}$.

Generator

An element $a$ of a group $G$ such that $\langle a \rangle = G$.

Order of an element

The least positive integer $m$ such that $a^m = e$, if such an integer exists.

Dihedral group

The group $D_n$ of all rotations and reflections of a regular $n$-gon, with $|D_n| = 2n$.

Rotation

An element $r^k$ of $D_n$ representing a turn of a regular $n$-gon by $\frac{2\pi k}{n}$ radians.

Reflection

An element $sr^k$ of $D_n$ with order $2$ that flips a regular $n$-gon across a symmetry axis.