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\section{Introduction}

This course will consist of 3 main sections:
\begin{itemize}
    \item Groups -- Continuation from IA,
        focussing on:
        \begin{itemize}
            \item Simple groups, $p$-groups,
                $p$-subgroups.
            \item Main result in this part of the
                course will be the Sylow theorems.
        \end{itemize}
    \item Rings -- Sets where you can add, subtract
        and multiply. For example
        \begin{itemize}
            \item $\ZZ$ or $\CC[X]$.
            \item Rings of integers $\ZZ[i]$,
                $\ZZ[\sqrt{2}]$ (more in part II
                number fields)
            \item Polynomial rings (Part II
                Algebraic Geometry)
        \end{itemize}
        A ring where you can divide is a field,
        for example $\QQ$, $\RR$, $\CC$ or
        $\ZZ/p\ZZ$ (prime $p$).
    \item Modules -- Analogue of vector spaces
        where the scalars belong to a ring instead
        of a field. We will classify modules over
        certain nice rings
        \begin{itemize}
            \item Allows us to prove Jordan Normal
                form and classify finite abelian
                groups.
        \end{itemize}
\end{itemize}

\newpage
\mychapter{Groups}

\newpage
\section{Revision and Basic Theory}

\begin{definition*}[Group]
    A group is a pair $(G, \cdot)$ where $G$ is a
    set and $\cdot \colon G \times G \to G$ is a binary
    operator satisfying:
    \begin{itemize}
        \item Associativity: $a \cdot (b \cdot c)
            = (a \cdot b) \cdot c ~~ \forall a, b,
            c \in G$.
        \item Identity: $\exists e \in G$ such
            that $e \cdot g = g \cdot e = g ~~
            \forall g \in G$.
        \item Inverses: $\forall g \in G ~
            \exists g^{-1} G$ such that $g \cdot
            g^{-1} = g^{-1} \cdot g = e$.
    \end{itemize}
\end{definition*}

\subsubsection*{Remarks}

\begin{enumerate}[(i)]
    \item In checking $\cdot$ is well-defined,
        need to check \emph{closure}, i.e. $a, b
        \in G \implies a \cdot b \in G$. (This is
        implicit in the notation $\cdot \colon G
        \times G \to G$).
    \item If using additive (multiplicative)
        notation, then often write 0 (or 1) for
        identity.
\end{enumerate}

\begin{definition*}[Subgroup]
    A subset $H \subset G$ is a subgroup (written
    $H \le G$) if $h \cdot h' \in H ~ \forall h,
    h' \in H$ and $(H, \cdot)$ is a group.
\end{definition*}

\begin{flashcard}[subgroup-trick]
\begin{remark*}
    A subset $H$ of $G$ is a subgroup if
    \cloze{$H$ is \fcemph{non-empty} and
    $a, b \in H \implies a \cdot b^{-1} \in H$.}
\end{remark*}
\end{flashcard}

\subsubsection*{Examples}

\begin{enumerate}[(i)]
    \item Additive groups $(\ZZ, +) \le (\QQ, +)
        \le (\RR, +)$.
    \item Cyclic and dihedral groups. $C_n =
        \text{cyclic group of order $n$}$, $D_{2n}
        = \text{symmetric of a regular $n$-gon}$.
    \item Abelian groups: those $(G, \cdot)$ such
        that
        \[ a \cdot b = b \cdot a ~~ \forall a, b
        \in G \]
    \item Symmetric and alternating groups
        \[ S_n = \text{all permutations of $\{1,
        \dots, n\}$} \]
        \[ A_n \le S_n \text{subgroup of even
        permutations} \]
    \item 
        \begin{flashcard}[quaternion-group]
        Quaternion group $Q_8 = \{\pm 1, \pm i,
        \pm j, \pm k\}$ with
        \[ \cloze{ij = k, \quad ji = -k, \quad i^2 = -1,
        \ldots} \]
        \end{flashcard}
    \item General and special linear groups.
        \begin{itemize}
            \item $\GL_n(\RR) = \{\text{$n \times
                n$ matrices over $\RR$ with
                $\det \neq 0$, and $\cdot$ is matrix
                multiplication.}\}$
            \item $\SL_n(\RR) \subset \GL_n(\RR)$
                subgroup of matrices with
                determinant $1$.
        \end{itemize}
\end{enumerate}

\begin{definition*}
    The (direct) product of groups $G$ and $H$ is
    the set $G \times H$ with operation
    \[ (g_1, h_1) \cdot (g_2, h_2) = (g_1g_2,
    h_1h_2) \]
\end{definition*}

\noindent
Let $H \le G$, the left cosets of $H$ in $G$ are
the sets $gH := \{gh \colon h \in H\}$ for $g \in
G$. These partition $G$, and each has the same
cardinality as $H$. Deduce

\begin{theorem}[Lagrange's Theorem]
    Let $G$ be a finite group and $H \le G$. Then
    $|G| = |H| \cdot [G : H]$ where $[G : H]$ is
    the number of left cosets of $H$ in $G$. $[G :
    H]$ is the index of $H$ in $G$.
\end{theorem}

\begin{remark*}
    Can also carry this out with right cosets.
    Lagrange $\implies$ $\text{number of left
    cosets} = \text{number of right cosets}$.
\end{remark*}

\begin{definition*}
    Let $g \in G$. If $\exists n \ge 1$ such that
    $g^n = 1$, then the least such $n$ is the
    order of $g$. Otherwise $g$ has infinite
    order.
\end{definition*}

\begin{remark*}
    If $g$ has order $d$, then
    \begin{enumerate}[(i)]
        \item $g^n = 1 \implies d \mid n$.
        \item $\{1, g, \dots, g^{d - 1}\} \le G$
            and so if $G$ is finite then $d \mid
            |G|$ (Lagrange).
    \end{enumerate}
\end{remark*}

\noindent
A subgroup $H \le G$ is normal if $g^{-1}Hg = H
~\forall g \in G$. We write $H \normalsub G$.

\begin{proposition}
    If $H \normalsub G$, then the set $G / J$ of
    left cosets of $H$ in $G$ is a group (called
    the quotient) with operation $g_1 H \cdot g_2
    H = g_1 g_2 H$.
\end{proposition}

\begin{proof}
    Check $\cdot$ well defined. Suppose $g_1 H =
    g_1' H$ and $g_2H = g_2'H$. Then $g_1' = g_1
    h_1$ and $g_2' = g_2h_2$ for some $h_1, h_2
    \in H$. Then
    \[ \implies g_1'g_2' = g_1h_1g_2h_2 = g_1 g_2
    \ub{(g_2^{-1} h_1 g_2)}_{\in H} \ub{h_2}_{\in H} \]
    \[ \implies g_1'g_2' H = g_1g_2 H \]
    Associativity is inherited from $G$, the
    identity is $H = eH$ and the inverse of $gH$
    is $g^{-1}H$.
\end{proof}

\begin{definition*}
    If $G$, $H$ are groups, a function $\phi : G
    \to H$ is a group homomorphism if
    \[ \phi(g_1 g_2) = \phi(g_1) \phi(g_2) ~
    \forall g_1, g_2 \in G \]
\end{definition*}

\noindent
It has kernel $\Ker(\phi) := \{g \in G \mid
\phi(g) = 1\} \le G$, and image $\Im(\phi) :=
\{\phi(g) \mid g \in G\} \le H$.

\myskip
If $a \in \Ker(\phi)$ and $g \in G$, then
\[ \phi(g^{-1}ag) = \phi(g^{-1}) \ub{\phi(a)}_{=1}
\phi(g) = 1 \]
so $g^{-1}ag \in \Ker(\phi)$. So $\Ker(\phi)
\normalsub G$.
