\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsthm,latexsym}
\usepackage[letterpaper,margin=1in]{geometry}
\newtheorem{theorem}{\textbf{Theorem}}

\begin{document}

\author{}
\date{}
\title{Writing Mathematics}


\maketitle

Since Dummit and Foote did not include a proof of the First Isomorphism
Theorem, I posted a proof on the class webpage. I will reproduce it here and
then use it to illustrate some points about how mathematics can be written for
clarity. I am advocating writing in a particular style that I am asking you to
use in the WIM project.

\begin{theorem}
  Let $\varphi : G \longrightarrow H$ be a surjective homomorphism. Let $K$ be
  the kernel of $\varphi$. Then $K$ is a normal subgroup and $H \cong G / K$.
\end{theorem}

{\textbf{Note:}} Dummit and Foote do not assume that $\varphi$ is
surjective, but this formulation is equivalent to what they prove. (Why?)

\begin{proof}
  The fact that $K$ is normal is Proposition~7 on page 82 of Dummit and Foote.
  Define a map $\varphi : G / K \longrightarrow H$ by $\theta (g K) = \varphi
  (g)$ for the coset $g K \in G / K$. Since there may be more than one choice
  for the coset representative $g$ we must check that this is well-defined.
  This means that if $g K = h K$ we must prove that $\varphi (g) = \varphi
  (h)$. Indeed, $g K = h K$ implies that $h^{- 1} g K = K$ so $h^{- 1} g \in
  K$. By definition of the kernel, this means that $\varphi (h^{- 1} g) = 1_H$
  so $\varphi (h)^{- 1} \varphi (g) = 1_H$ or $\varphi (h) = \varphi (g)$,
  proving that $\theta$ is well-defined.
  
  Now we want to prove that $\theta$ is an isomorphism. First, we check that
  it is a homomorphism. If $g, h \in G$ then $g K \cdot h K = g h K$ so
  $\theta (g K \cdot h k) = \varphi (g h K) = \varphi (g h) = \varphi (g)
  \varphi (h)$ which equals $\theta (g K) \theta (h K)$, proving that
  $\varphi$ is a group homomorphism.
  
  Now we ject that $\theta$ is an isomorphism. It is surjective because we are
  assuming that $\varphi$ is surjective, so every element of $H$ equals
  $\varphi (g) = \theta (g K)$ for some $g \in G$. To show that it is
  injective we just reverse the argument that it is well-defined: if $\theta
  (g K) = \theta (h K)$ then $\varphi (g) = \varphi (h)$, so $g^{- 1} h$ is in
  the kernel $K$ of $\varphi$. Therefore $g^{- 1} h K = K$ so $g K = h K$.
  Thus $\theta$ is a bijective homomorphism, that is, an isomorphism.
\end{proof}

Now let me explain the reasons that we write the proof this way, and to
characterize this style of writing.

\begin{itemize}
   \item Encapsulate anything technical in a proof. 
\end{itemize}

In this style
of writing, the text consists of a series of definitions, results (with
proofs) punctuated by less technical comments intended and inform to guide the
reader. To fully absorb the material, the reader must understand how
everything is proved, but they can obtain an overview by skipping the proofs,
then returning to them in a second reading. Anything that is technical or
difficult will slow the reader down. So these details are put into proofs and
never interjected in the middle of an informal discussion.

On the other hand the material that is {\textit{not}} put into proofs can
contain references to the literature, facts you are not going to prove,
informal restatements intended to convey intuition, examples and remarks.

\section{{\LaTeX}}

It is best to use {\LaTeX} to prepare your documents. Among LaTeX's features
are automatic numbering of definitions, equations, results, remarks, etc. Some
text is rendered in {\textit{math mode}}.

Decide which are your most important results. These are called
{\textit{theorems}}. A first reading of a paper may consist of just absorbing
the contents of these results. Less important facts are called
{\textit{propositions}}. A {\textit{lemma}} is an auxiliary fact that doesn't have
significance for itself, but which is stated separately either because it will
be used more than once, or because it aids the narrative flow to isolate it
before presenting the proof of a theorem or proposition. A {\textit{corollary}}
is an important fact that follows immediately from a theorem. A
{\textit{remark}} is some observation or informational text that you want to
refer to later.

The appendix below contains some practical guidance for using LaTeX.

\section{Stylistic considerations in a proof}

\begin{itemize}
  \item Before writing, decide who your intended audience is.
\end{itemize}
Depending on your intended audience, you may adopt a style that is very brief
(omitting details that the reader can supply themselves) or more expansive
(including more details). A condensed style is appropriate for a journal, and
a more expansive style is more appropriate for an expository note or for a
book.

Homework is a special case: in writing homework solutions the presumption is
that the reader knows how to solve the problem, and your task is to convince
them that {\textit{you}} understand. \textit{That is not the type of writing we are
talking about here.} Assume that you know something the reader does not, and
your task is how best to convey that.

\begin{itemize}
  \item Be able to put yourself in the reader's place.
\end{itemize}
The key to good writing is the ability to look at a piece of text the same way
your intended audience will look at it. This is a habit that can be
cultivated. You can anticipate their difficulties. For example, you might
be stating some fact that is important but if you are not careful about
the narrative the reader might not understand \textit{why} you are stating it. 
If you are able to put yourself in the reader's place you can avoid such pitfalls.

\begin{itemize}
  \item ``Less is more.''
\end{itemize}
Adding more detail to a text can help inexperienced readers, but it always
comes at a price. Giving details the reader already knows slows them down and
makes the key points harder to grasp. The best writing strives for
succinctness and clarity. Too much detail is always bad.

\begin{itemize}
	\item Defined terms should be italicized.
\end{itemize}

There is a formal LaTeX environment for definitions, which can
be put in an environment like this:

\begin{verbatim}
\begin{definition}
	\label{def:normal}
	A subgroup $H$ of $G$ is called \textit{normal} if $gHg^{-1}=H$ for all
	$g\in G$.
\end{definition}
\end{verbatim}

But you don't have to use this environment: you can just put the definition in
the text, (Use the formal environment if for some reason you will want to refer
to the definition later.) In any case remember to italicize the defined term.

\

Since we have advocated a style of writing that separates more technical
parts into proof of statements (theorems, propositions or lemmas) we will now
focus on how to write the proof.

\begin{itemize}
  \item Use paragraphs to delineate the main points.
\end{itemize}
Figure out what the key points you want to communicate are, and separate them
into paragraphs. The above proof defines a map $\theta : G / K \longrightarrow
H$, and establishes three main points, each deserving a paragraph. These are
(1) the map $\theta$ is well-defined, (2) the map $\theta$ is a group
homomorphism and (3) the map $\theta$ is bijective. Each gets its own
paragraph..
\begin{itemize}
  \item Use narrative devices to make sure the reader always knows where you
  are going with an argument and why.
\end{itemize}
For example, we wrote:

\begin{quotation}
  Since there may be more than one choice for the coset representative $g$ we
  must check that this is well-defined. This means that if $g K = h K$ we must
  prove that $\varphi (g) = \varphi (h)$. Indeed, $g K = h K$ implies that
  $h^{- 1} g K = K$ so $h^{- 1} g \in K$. By definition of the kernel, this
  means that $\varphi (h^{- 1} g) = 1_H$ so $\varphi (h)^{- 1} \varphi (g) =
  1_H$ or $\varphi (h) = \varphi (g)$, proving that $\theta$ is well-defined.
\end{quotation}

This starts by telling the reader what the issue is, then paraphrases
well-definedness as a concrete assertion, that $g K = h K$ implies $\varphi
(g) = \varphi (h)$. Then the word ``indeed'' is used to carry the reader
forward, indicating that the next bit will establish this point. Other phrases
such as ``to see this'' or ``this is because'' can serve the same purpose.
\begin{itemize}
  \item Read what you have written.
\end{itemize}
Once you have written your text, read it. Try to put yourself in the position
of a reader who is trying to follow the argument, and see if the ideas are
clear.

\section*{Appendix: LaTeX Hints}

At the class web page I've posted a sample LaTeX document called
\texttt{normal.tex} that you can copy or study.

Every LateX document has some preliminary setup material at
the beginning called the \textit{preamble}. This is the preamble for
\texttt{normal.tex}:

\begin{verbatim}
\documentclass[12pt]{article}
\usepackage{amsmath, amssymb, amsthm}
\usepackage[letterpaper,margin=1in]{geometry}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{theorem}{Theorem}
\end{verbatim}

The first line tells us to use the article document class
and to use 12pt fonts. The default font size is 10pt but we
recommend using 12pt for most writing. 

After the documentclass is loaded, some packages may be included.
In this example the AMS (American Math Society) packages \texttt{amsmath},
\texttt{amssymb} and \texttt{amsthm} packages are loaded. These give a useful
framework for writing. The \texttt{amssymb} package gives you a useful set of
symbols, and \texttt{amsthm} gives a framework for definitions, theorems, and
propositions which can be declared as in the \texttt{newtheorem} lines. 

In LaTeX there are provisions for automatic numbering of
Theorems, Propositions, equations, etc.
Our discussion will assume that your latex preamble
includes the amsymb, amsmath and amsthm packages.
If your document contains a Proposition, you need to declare
it here, etc.

If you want Theorems, Propositions and Lemmas,
numbered in a single sequence, you may alternatively do this:

\begin{verbatim}
\newtheorem{theorem}{\textbf{Theorem}}
\newtheorem{proposition}[theorem]{\textbf{Proposition}}
\newtheorem{lemma}[theorem]{\textbf{Lemma}}
\end{verbatim}

The body of the document begins with a line

\begin{verbatim}
\begin{document}
\end{verbatim}

To make use of the automatic numbering, textual elements are
assigned labels, and may then be referred to using references.

\begin{verbatim}
\begin{theorem}
\label{thm:kernorm}
Let $\phi:G\to H$ be a homomorphism. Then the kernel
\begin{equation}
\label{eq:kerdef}
\ker(\phi)=\{x\in G|\phi(x)=1\}
\end{equation}
is a normal subgroup of $G$.
\end{theorem}
\end{verbatim}

This contains two labels, one for the Theorem (labeled \texttt{thm:kernorm})
and one for the equation (labeled \texttt{eq:kerdef}). This text produces a theorem
containing a numbered equation:

\begin{theorem}
\label{thm:kernorm}
Let $\phi:G\to H$ be a homomorphism. Then the kernel
\begin{equation}
\label{eq:kerdef}
\ker(\phi)=\{x\in G|\phi(x)=1\}
\end{equation}
is a normal subgroup of $G$.
\end{theorem}

Later you may reference these using the labels:

\begin{verbatim}
Using Theorem \ref{thm:kernorm} we see that $\mathop{ker}(\phi)$
defined by \eqref{eq:kerdef} is a normal subgroup.
\end{verbatim}

This supplies the labels automatically. 

\begin{itemize}
	\item When you tex the document run latex \textit{twice}.
\end{itemize}

The first pass creates an auxiliary file with the \texttt{.aux} suffix. The
second time, the labels are inserted:

\medbreak
Using Theorem \ref{thm:kernorm} we see that $\mathop{ker}(\phi)$
defined by \eqref{eq:kerdef} is a normal subgroup.

\medbreak
This scheme is robust in that if you move text around
and the numbering changes, the labels will remain correct.

\end{document}
