\section{Groups (review)}

\definition{}
Before we continue, we must introduce a bit of notation:
\begin{itemize}
	\item $S_n$ is the set of permutations on $n$ objects.
	\item $\mathbb{Z}_n$ is the set of integers mod $n$.

	\item $\mathbb{Z}_n^\times$ is the set of integers mod $n$ with multiplicative inverses. \par
	In other words, it is the set of integers smaller than $n$ and coprime to $n$.\footnotemark{} \par
	For example, $\mathbb{Z}_{12}^\times = \{1, 5, 7, 11\}$.

	\footnotetext{We proved this in another handout, but you may take it as fact here.}
\end{itemize}

\problem{}
What are the elements of $S_3$? \tab\hint{Use cycle notation}\par
How about $\mathbb{Z}_{17}^\times$?

\vfill


\definition{}
A \textit{group} $(G, \ast)$ consists of a set $G$ and an operator $\ast$. \par
Groups always have the following properties:

\begin{enumerate}
	\item $G$ is closed under $\ast$. In other words, $a, b \in G \implies a \ast b \in G$.
	\item $\ast$ is \textit{associative}: $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a,b,c \in G$
	\item There is an \textit{identity} $e \in G$, so that $a \ast e = a \ast e = a$ for all $a \in G$.
	\item For any $a \in G$, there exists a $b \in G$ so that $a \ast b = b \ast a = e$. $b$ is called the \textit{inverse} of $a$. \par
	This element is written as $-a$ if our operator is addition and $a^{-1}$ otherwise.
\end{enumerate}

Any pair $(G, \ast)$ that satisfies these properties is a group.

\problem{}
Is $(\mathbb{Z}_5, +)$ a group? \par
Is $(\mathbb{Z}_5, -)$ a group? \par
\note[Note]{$+$ and $-$ refer to the usual operations in modular arithmetic.}
\vfill


\problem{}
What is the group with the fewest elements?

\begin{solution}
	Let $(G, \star)$ be our group, where $G = \{x\}$ and $\star$ is defined by $x \star x = x$

	Verifying that the trivial group is a group is trivial.
\end{solution}

\vfill
\pagebreak











\problem{}
Show that function composition is associative

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\problem{}
Show that $S_n$ is a group under composition.

\vfill

\problem{}
Let $(G, \ast)$ be a group with finitely many elements, and let $a \in G$. \par
Show that $\exists n \in \mathbb{Z}^+$ so that $a^n = e$ \par
\hint{$a^n = a \ast a \ast ... \ast a$ repeated $n$ times.}

\vspace{2mm}

The smallest such $n$ defines the \textit{order} of $g$.

\begin{examplesolution}
	We've already done a special case of this problem! \par
	Find it in this handout, then rewrite your proof for an arbitrary (finite) group.
\end{examplesolution}


\vfill

\problem{}
What is the order of 5 in $(\mathbb{Z}_{25}, +)$? \par
What is the order of 2 in $(\mathbb{Z}_{17}^\times, \times)$? \par

\vfill
\pagebreak














\definition{}<gendef>
Let $G$ be a group, and let $g$ be an element of $G$. \par
We say $g$ is a \textit{generator} if every other element of $G$ may be written as a power of $g$. \par

\problem{}
Say the size of a group $G$ is $n$. \par
If $g$ is a generator, what is its order? \par
Provide a proof.
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\problem{}
Find the two generators in $(\mathbb{Z}, +)$ \par
Then, find all generators of $(\mathbb{Z}_5, +)$
\vfill


\problem{}
How many groups have only one generator?

\begin{solution}
	Only one: the trivial group. The inverse of a generator is also a generator!
\end{solution}

\vfill


\definition{}
Let $S$ be a subset of the elements in $G$. \par
We say that $S$ \textit{generates} $G$ if every element of $G$ may be written as a product of elements in $S$. \par
\note{Note that this is an extension of \ref{gendef}.}

\problem{}
We've already found a few generating sets of $S_n$. What are they?

\begin{solution}
	The following sets generate $S_n$:
	\begin{itemize}
		\item All transpositions
		\item All transpositions of the form $(1, k)$
		\item All adjacent transpositions
	\end{itemize}

	\vspace{2mm}

	The smallest generating set of $S_n$ consists of the transposition $(12)$ and the $n$-cycle $(1,2,...,n)$. \par
	The proof of this is a bonus problem later in the handout.
\end{solution}

\vfill
\pagebreak
