\chapter{Cyclic Groups}



Sometimes it is interesting to see how many distinct elements can be formed by repeating the operation on some specific element; we call this the order of the element.



\section{Cyclic Groups}

\begin{definition}[Cyclic group]
A \emph{cyclic group} $(G,\circ)$ is a group such that there exists some element $x$ such that all elements are of the form $x^n$. We say that such a $x$ \emph{generates} $G$.
\[ G \text{ is cyclic } \iff \exists x \in G [ \forall g \in G ( \exists n \in \mathbb{Z} [ x^n = g   ] ) ] \]
Such an $x$ is called a \emph{generator of $G$}
\end{definition}


If the group operation is clear, one may write the set of elements generated by the element $x$ as $\langle x \rangle$.
\[ \langle x \rangle = \{ x^{n} : n \in \mathbb{Z} \} \].

One may see that this is indeed forms its own group, and furthermore, a cyclic one.
\begin{proposition}
\[(G,\circ) \land x \in G\]
\[  ( \langle x \rangle, \circ) \leq (G,\circ )  \]
\[ ( \langle x \rangle, \circ) \text{ is cyclic} \]
\end{proposition}

This is how we construct cyclic subgroups, and this gives us a reason to think of 'orders of elements' as 'orders of the cyclic subgroup they generate'.
\begin{definition}[Order of a group element]
The \emph{order} of a group element $g$ is the cardinality of $\langle g \rangle$.
The order of $g$ is denoted as $\mathrm{ord}(g)$.
\[ \mathrm{ord}(g) = | \langle g \rangle | \]
\end{definition}





\section{Isomorphisms of Cyclic Groups}


Finite cyclic groups of the same order are all isomorphic by the isomorphism $ f ( \gamma^n )= \eta^n$, where $ G = \langle \gamma \rangle , H = \langle \eta \rangle$, hence we define a notation to represent the unique cyclic group of order $n$ (unique up to an isomorphism). 

\begin{example}
The \emph{cyclic group of order $n$} $\mathrm{Cyc}(n)$
Cyclic group of $n$ elements.
$\mathrm{Cyc}(n)= \langle x : x^n = 1 \rangle$
\end{example}

When proving theorems on cyclic groups, we often find ourselves working in the exponent of the same element, and the exponent acts like modular arithmetic; the law $g^{a}\circ g^{b} = g^{a+b}$ looks and feels very much like an isomorphism to $\mathbb{Z} / n \mathbb{Z}$ (where $n$ is the order of $g$). This suspicion is correct, and leads us to one of the most powerful methods of thinking of cyclic groups; treating them like modular arithmetic.

\begin{proposition}
$f : \mathrm{Cyc}(n) \to \mathbb{Z} / n\mathbb{Z}$
$f(x^n)=[n]$
\[ \mathrm{Cyc}(n) \cong \mathbb{Z} / n\mathbb{Z} \]
\end{proposition}


\section{Properties of Cyclic Groups}


\begin{proposition}
Groups of prime order are cyclic.
\[ |G| \text{ is prime } \implies G \cong \mathrm{Cyc}(|G|) \]
\end{proposition}
Lagrange's theorem implies that groups of prime order only have themselves and the trivial group as subgroups, hence each element is a generator of the group, hence it is cyclic.


The isomorphic equivalence $\mathrm{Cyc}(n) \cong \mathbb{Z} / n\mathbb{Z}$ really means that cylic groups are actually all about additive integer groups modulo $n$. This permits us to invoke techniques from number theory to fully realize the properties of cyclic groups.

The following lemma helps us understand more about the cyclic nature of $\mathbb{Z}/n\mathbb{Z}$

\begin{proposition}
Let $a \in \mathbb{Z} /n \mathbb{Z}$, then $\langle a \rangle = \mathbb{Z} / n\mathbb{Z}$ iff $a$ is coprime to $n$.
\end{proposition}

\begin{corollary}
There are $\varphi(n)$ generators of $\mathrm{Cyc}(n)$.
\end{corollary}


Here is a more constructuve result which calculates the order of any element of $\mathbb{Z}/n\mathbb{Z}$.
\begin{corollary}
Let $x \in \mathbb{Z} /n \mathbb{Z}$, then the order of $x$ is $\frac{ \mathrm{lcm} (x,n)}{x}$, or equivalently, $\frac{n}{\gcd(x,n)}$.
\end{corollary}


\begin{corollary}
$a \in \mathbb{Z} /n \mathbb{Z}$ is a generator of the group iff $\gcd(a,n)=1$.
\end{corollary}




Now to prove some propositions on the subgroups of cyclic groups.

\begin{proposition}
Subgroups of cyclic groups are cyclic.
\end{proposition}
The following proposition is proved by switching from $\mathrm{Cyc}(n)$ to $\mathbb{Z} /n\mathbb{Z}$ showing that all elements in subgroups of $\mathbb{Z} / n\mathbb{Z}$ must be multiples (repeated operation) of the smallest nonzero number in that subgroup (if such a smallest nonzero number is in the group, which is true for all but the trivial group). This is a good example of leveraging the properties of integers, bringing their nice properties to facilitate proofs.




\begin{theorem}[Fundamental theorem of cyclic groups]
If $d$ divides the order of a cyclic group, then there is a unique cyclic subgroup of order $d$.
\[  d | n \implies \exists! H \leq \mathrm{Cyc}(n) [|H|=d]\]
\end{theorem}
Again leveraging number theory, existence follows by noting that $\frac{|G|}/d$ generates a group isomorphic to $\mathbb{Z} / d \mathbb{Z}$ (which is cyclic). Then the proof is finished by showing that any other generator of order $d$ is a multiple of our $\frac{n}{d}$.


Just as modular groups can be used to prove stuff about cyclic groups, the theory of cyclic groups can conversely prove propositions in number theory.

\begin{proposition}
\[ n = \sum_{d|n} \varphi(d) \]
\end{proposition}