Math322_Notes.tex

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\title{Math 322 Notes}
\author{Henry Xia}
%\date{15 September 2017}

\begin{document}

\maketitle

\tableofcontents

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\section{Introduction}

\begin{defn}
	A \emph{quotient set} is a set $S/\sim$ whose elements are in one to one correspondence
	with equivalence classes of $S$. We also write $\overline{S} = S/\sim$.
\end{defn}

\begin{defn}
	A \emph{natural map} $S\to S/\sim$ is a surjective map to the equivalence classes of
	$S$. 
\end{defn}


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\begin{defn}
	A \emph{semigroup} is a set $S$ that is closed under multiplication. That is $\forall
	a,b\in S$, we have $ab\in S$. 
\end{defn}

\begin{defn}
	A \emph{monoid} is a semigroup that has an identity element. That is $\exists 1\in S$ such
	that $1a = a1 = a$ for all $a\in S$. 
\end{defn}

\begin{defn}
	A \emph{group} is a monoid where every element has an inverse. That is $\forall a\in S$,
	there exists some $a^{-1}$ such that $aa^{-1} = 1$. 
\end{defn}

\begin{defn}
	A \emph{subgroup} is a subset of a group that is also a group.
\end{defn}

\begin{defn}
	An \emph{Abelian group} is a group whose multiplication is commutative. 
\end{defn}

\begin{defn}
	The \emph{symmetric group} on $n$ elements is the set of all permutations of $n$
	elements. We denote this as $S_n$. 
\end{defn}

\begin{defn}
	A \emph{cyclic group} is a group that can be generated by one of its elements. That is
	$G = \set{a, a^2, \dots, a^{n-1}, a^n=1}$. We say that $a$ generates $G$.
\end{defn}

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\begin{defn}
	The \emph{order} of a group is the number of elements in the cardinality of the group.
	The order of an element $a$ is the smallest $m$ such that $a^m = 1$. If no such $m$
	exists, we say $a$ has infinite order. This is equivalently the order of the group
	generated by $a$. 
\end{defn}

\begin{defn}
	The \em{direct product} is the cartesian product of the groups, where the
	group action is defined componentwise.
\end{defn}

\begin{defn}
	We say a group $G$ is isomorphic to a group $H$, or $G\simeq H$, if there exists a
	bijection from $G$ to $H$ that preserves the group action. That is there exists some
	bijection $f:G\to H$ such that $f(xy) = f(x)f(y)$.
\end{defn}

\begin{example}
	The group $\RR$ with addition is isomorphic to $\RR^*_+$ with multiplication. We can
	take $f(x)=e^x$, then $f(x+y) = e^{x+y} = e^x e^y = f(x)f(y)$. 
\end{example}

\begin{thm} [Cayley's Theorem]
	Any finite group $G$ is isomorphic to a subgroup of the symmetric group acting on $G$.
\end{thm}
\proof
Let $G = \set{x_1,x_2,\dots,x_n}$ have order $n$. Then for each $a\in G$, define
$f_a:G\to G$ where $f_a(x) = ax$. We claim that $f_a$ is a bijection. It suffices to
show that $f_a$ is an injection. Suppose that $f_a(x) = f_a(y)$, then $ax=ay \implies
a^{-1}ax=a^{-1}ay \implies x=y$.

Let $\phi:G\to S_n$ map each element $a\in G$ to the element of $S_n$ that corresponds to
$f_a$. Now we need to check that $\phi$ is injective. Indeed, if $\phi(a) = \phi(b)$, then
$ax = f_a(x) = f_b(x) = bx \implies a=b$. We also need to check that $\phi(ab) =
\phi(a)\phi(b)$. Indeed, $\phi(ab)$ maps $x$ to $abx$, $\phi(b)$ maps $x$ to $bx$, and
$\phi(a)$ maps $bx$ to $abx$.
\qedhere

\begin{example}
	Is $(\QQ,+)$ isomorphic to $(\QQ^*_+,\times)$?

	No. Consider $2x=a$, where $a\in\QQ$. There exists some $x\in\QQ$ for all $a$. This 
	equation becomes $f(x)^2=f(a)$ should the two groups be isomorphic, however, it is
	clear that $f(x)$ does not exist for all $f(a)$. 
\end{example}

\begin{example}
	Fix $a\in G$, then let $C = \set{b\in G : ab=ba}$. We call $a$ the centralizer.
	Observe that $C$ is a subgroup of $G$. 

	It is obvious that $1\in C$.

	Let $x,y\in C$, then $xya = xay = axy$ by associativity, hence $xy\in C$. 

	Let $x\in C$, then $x^{-1}a = x^{-1}axx^{-1} = x^{-1}xax^{-1} = ax^{-1}$ by
	associativity, hence $x^{-1}\in C$. 
\end{example}

\begin{defn}
	The \emph{center} of a group $G$ is the subgroup $\set{a:ax=xa ~~\forall x\in G}$. 
\end{defn}

\begin{remark}
	The intersection of subgroups is also a subgroup. 
\end{remark}

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