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\begin{document}


\noindent
    {\bf\large
      What everyone already knows about groups.} \\
    (Dummit \& Foote Chapters 1-9, approximately.)

\medskip

\begin{enumerate}
\item
  Definitions. Language. Terminology, like `order of a group
  or element', `exponent of a group', `index of a subgroup'.
  Notation, like $|G|$, $G/N$, $[G:H]$, $[x,y]$.
  Easy syntactic results, like: the group axioms imply
  (i) $(x^{-1})^{-1}=x$ and (ii) $(xy)^{-1}=y^{-1}x^{-1}$.
  
\item
  Examples:
  \begin{enumerate}
    \item
  $\mathbb Z$, $\mathbb Z_n$, $S_n$, $A_n$, $D_n$, $D_{\infty}$,
  $Q_8$, ${\textrm {GL}}_n(\mathbb F)$,
      ${\textrm {SL}}_n(\mathbb F)$,
${\textrm {PSL}}_n(\mathbb F)$,
      ${\textrm O}_n(\mathbb F)$.
  \begin{itemize}      
\item Know how to compute in these groups, so know
  modular arithmetic, cycle decomposition of permutations,
  matrix arithmetic.
  \end{itemize}  
\item
  Groups of small order: know a complete list of groups of order $<16$.
\end{enumerate}
    
\item
  H, S, P:
  \begin{enumerate}
  \item  Homomorphisms.  Inner/outer automorphisms.
  \item Subgroups/normal subgroups. Subgroups of index 2 are normal.
  \item Quotient groups.
  \item Lattices of subgroups/normal subgroups.
  \item
    Left invariant equivalence relations on groups. Cosets.
  Double cosets.
\item Characterization of products and of semidirect products. 
  \end{enumerate}
    
\item
Basic structure theorems:
\begin{enumerate}
\item Cayley Representation Theorem.
\item Noether's Isomorphism Theorems. (3)
\item Correspondence Theorem.
\item Jordan-H\"{o}lder Theorem.
\item Lagrange's Theorem.
\item Cauchy's Theorem.
\item Sylow's Theorems.
\end{enumerate}
  
  
\item
Commutator theory:
\begin{enumerate}
\item Abelianness.
\item Centralizers.
\item Nilpotence: center, ascending/descending central series.
\item solvability: derived group/series.
\end{enumerate}
    
\item
Group actions:
\begin{enumerate}
\item Equivalent definitions, terminology ($G$-set,
  orbit, transitive action,
  (semi)regular action, homomorphisms/automorphisms).
\item Orbit-Stabilizer Theorem.
\item Class equation.
\item Structure Theorem.
\item Conjugacy action on elements, subgroups. Conjugacy
  classes, normalizers. Conjugacy classes in $S_n$, $A_n$,
  ${\textrm{GL}}_n(\mathbb F)$.
\end{enumerate}

\item
Abelian groups:
\begin{enumerate}
\item Structure Theorem for f.g. abelian groups.
\item Invariant factor form versus elementary divisor form.
\end{enumerate}


\item
Infinite groups:
\begin{enumerate}
\item Free groups, reduced words.
\item Presentations. Universal property of presentations.
\item Coproducts of groups given by presentations.
\end{enumerate}
  
\end{enumerate}

\vfill
\pagebreak


\noindent
{\large \bf Terminology for functions.}

\medskip
Let $A$ and $B$ be sets and let $f\colon A\to B$ be a function
from $A$ to $B$. There are sets and functions related to $A, B$ and $f$
that have special names.

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\medskip

\begin{enumerate}
\item
The 
{\it image} of $f$ is 
$\image(f) = f[A] = \{b\in B\;:\;\exists a\in A(f(a)=b)\}$.
The image of a subset $U\subseteq A$ is
$f[U] = \{b\in B\;:\;\exists u\in U(f(u)=b)\}$.
\item
The {\it preimage} or {\it inverse image}
of a subset $V\subseteq B$ is
$f^{-1}[V] = \{a\in A\;:\;f(a)\in V\}$.
\item
The preimage of a singleton $\{b\}$ is written 
$f^{-1}(b)$ and sometimes called the 
{\it fiber} of $f$ {\it over} $b$.
The fiber $f^{-1}(f(a))$ containing the element $a$ is sometimes
written $[a]$.
\item
The {\it kernel} of $f$ is
$\ker(f)=\{(a,a')\in A^2\;:\;f(a)=f(a')\}$, which is an equivalence
relation on $A$.
\item
  The {\it kernel class} of an element $a\in A$
  is the same as the fiber $[a]$ that contains it.
  It is often written $a/\ker(f)$ to emphasize that
  it is a kernel class.
\item
The {\it coimage} of $f$ is the set 
$$\coim(f)=\{f^{-1}(b)\;:\;b\in \image(f)\}=\{[a]\;:\;a\in A\}=A/\ker(f)$$
of all nonempty fibers. 
\item
The {\it natural map} is
$\nu\colon A\to \coim(f)\colon a\mapsto [a]$.
(More generally, for any equivalence relation $E$ on $A$, 
the natural map is
$\nu\colon A\to A/E\colon a\mapsto a/E$.)
\item
The {\it inclusion map} is 
$\iota\colon \image(f)\to B\colon b\mapsto b$.
(More generally, for any subset $S$ of $B$, 
the inclusion map is
$\iota\colon A\to B\colon b\mapsto b$.)
\item
The {\it induced map} is 
$\bar{f}\colon \coim(f)\to \image(f)\colon [a]\mapsto f(a)$.
\end{enumerate}

\medskip

Some facts:

\begin{enumerate}
\item
The natural map is {\it surjective}.
\item
The inclusion map is {\it injective}.
\item
The induced map is {\it bijective}.
\item
$f = \iota\circ \bar{f}\circ \nu$. (This is the
{\it canonical factorization} of $f$.)
%\item
%$A/\ker(f)$ is another notation for $\coim(f)$.
\end{enumerate}

\end{document}
