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Makefile

@@ -1,5 +1,5 @@
 BUILD_DIR:=.build
-LATEXRUN:=latexrun --latex-cmd=lualatex --latex-args="-shell-escape" -W no-all -O $(BUILD_DIR)
+LATEXRUN:=latexrun --latex-cmd=lualatex --latex-args="-shell-escape" -W no-overfull -O $(BUILD_DIR)
 PDFLATEX:=cd $(BUILD_DIR) && pdflatex -shell-escape -file-line-error
 LUALATEX:=cd $(BUILD_DIR) && lualatex -shell-escape -file-line-error
 RUBBER:=cd $(BUILD_DIR) && rubber -d --shell-escape

abstract-algebra.tex

@@ -2,16 +2,11 @@
 
 \section{Thoughts}
 
-\subsection{0}
-
-I think that the elements of a group should typically be thought of as functions, with the group operation
+\begin{itemize}
+\item It seems that the elements of a group can be thought of both as permutations/mappings, and also ``configurations​'' or ``states​'' of a certain system. The correspondence is that a given mapping is the mapping from an agreed initial​ state to its corresponding state.
+\item I think that the elements of a group should typically be thought of as functions, with the group operation
 being composition.
-
-
-
-
-\subsection{1}
-The elements of a group represent permutations, and the group operation is composition.
+\item The elements of a group represent permutations, and the group operation is composition.
 
 So what is a group under addition? Do its elements also represent permutations?
 
@@ -20,6 +15,9 @@ \subsection{1}
 So we can certainly think of the elements as permutations and addition as composition.
 
 What about $\R^\times$? I think we can think of that as a permutation of $\R$ also.
+\end{itemize}
+
+
 
 
 
@@ -411,6 +409,19 @@ \section{Groups}
   The order of $\gamma$ is 6.
 \end{example*}
 
+\begin{theorem*}
+  The order of a permutation is equal to the $\lcm$ of the lengths (i.e. orders) of the cycles in
+  its cycle decomposition.
+\end{theorem*}
+
+\begin{intuition*}
+  The first time we hit the identity is the first time that all component cycles hit identity
+  simultaneously. A given $m$-cycle will hit identity every $m$ iterations. Therefore the first
+  iteration at which the overall permutation hits identity is the first iteration that's a multiple
+  of all the component cycle orders.
+\end{intuition*}
+
+
 \section{Examples of groups, homomorphisms and quotients}
 
 \subsection{Finite order}

foundations.tex

@@ -310,38 +310,93 @@ \subsection*{\url{https://www.reddit.com/r/explainlikeimfive/comments/61siyw/eli
 not as common as the standard trig functions.
 
 
+
 \begin{definition*}
 
- $c | a$, i.e. $c$ is a \defn{divisor} of $a$ (and $a$ is a \defn{multiple} of $c$), if there exists $d$ such that $cd = a$.
 
- $\gcd(a, b)$, the \defn{greatest common denominator} of $a$ and $b$, is the largest integer $c$ such that $c | a$ and $c | b$.
+  \defn{prime factorization}
+  \begin{mdframed}
+    \includegraphics[width=400pt]{img/foundations--integers-e251.png}
+  \end{mdframed}
+
+ $c | a$, i.e. $c$ \defn{divides} $a$ ($a$ is a \defn{multiple} of $c$), i.e. if there exists $d$ such that $cd = a$.
+
+ $\gcd(a, b)$, the \defn{greatest common divisor} of $a$ and $b$, is the largest integer $c$ such that $c | a$ and $c | b$.
 
   $a$ and $b$ are \defn{relatively prime} aka \defn{coprime} if $\gcd(a, b) = 1$.
 
   $\lcm(c, d)$ is the smallest integer $a$ such that $c|a$ and $d|a$.
 \end{definition*}
 
+  Question: why exactly are we contemplating products here, as opposed to e.g. sums?
+
+
+
+
 \begin{remark*}
+  a \defn{multiple} of a prime is a number whose factorization contains that prime.
+
+  $c$ \defn{divides} $a$ ($a$ is a \defn{multiple} of $c$) if $a$'s factorization is a ``multiplicity superset​'' of $c$'s.
+
   The $\gcd$ is the largest integer whose factorization is a ``subset​'' of both factorizations.
 
-  If $a$ and $b$ are relatively prime then their factorizations have no primes in common.
+  If $a$ and $b$ are relatively prime then their factorizations have no overlap.
 
   The $\lcm$ is the smallest integer whose factorization is a ``superset​'' of both factorizations.
 
-  For example if
+  If $a$ is a multiple of $c$ then their $\lcm$ is $a$.
+
+  If $a$ and $b$ are relatively prime then their $\lcm$ is their product.
+\end{remark*}
+
+\begin{example*}
+Consider $2^2 \cdot 3 = 12$ and $2\cdot 3^2 = 18$.
+
+Their factorizations have much in common (they are certainly not coprime), but neither is a
+multiple of the other.
+
+Their $\lcm$ is the first number that is a ``multiplicity superset​'' of the other: i.e. we must add
+a factor of $3$ to $12$'s factorization, or an additional $2$ to $18$'s, either way
+yielding $2^2 \cdot 3^2 = 36$.
+
+Note that their $\lcm$ is not one of them (as it would be if one divided the other), but it is
+smaller than their product.
+
+The $\gcd$ of $2^2\cdot 3$ and $2\cdot 3^2$ is $2\cdot 3$.
+\end{example*}
+
+
+\begin{example*}
+    For example if
   \begin{align*}
     a = 12 &= 2^2 \cdot 3 \\
     b = 40 &= 2^3 \cdot 5 \\
   \end{align*}
   then $\gcd(a, b) = 2^2 = 4$ and $\lcm(a, b) = 2^3 \cdot 3 \cdot 5 = 120$.
 
-  This can be written as a general theorem involving mins and maxes in the exponents of a product of primes.
-\end{remark*}
+  This can be written as a general theorem involving mins and maxes in the exponents of a product
+  of primes.
+\end{example*}
+
 
 \begin{theorem*}
   $\gcd(a,b) \times \lcm(a,b) = ab$
 \end{theorem*}
 
+\begin{example*}
+  The product of  $2^2\cdot 3$ and $2\cdot 3^2$ is their concatenation: $2^2 \cdot 3 \cdot 2 \cdot 3^2 = 12 \cdot 18 = 216$.
+
+  This is the product of their $\gcd$ $2\cdot 3$ and their $\lcm$ $2^2\cdot 3^2$.
+\end{example*}
+
+
+\begin{intuition*}
+  We're performing additive operations on the exponents of the prime factors. The product is the
+  sum of all; the $\gcd$ takes the min from each and is thus missing the maxes; the $\lcm$ takes
+  the max from each and is thus missing the mins; their product contains the full multiplicities of
+  all factors.
+\end{intuition*}
+
 
 \section{$\sqrt{2}$ is irrational}
 \begin{theorem*}

mathematics.sty

@@ -33,7 +33,6 @@
 \usepackage{graphicx}
 \usepackage[colorlinks=true,urlcolor=blue]{hyperref}
 \usepackage[T1]{fontenc}
-\usepackage[utf8]{inputenc}
 % \usepackage{mathabx}
 \usepackage{mathrsfs}
 \usepackage{mathtools}
@@ -444,4 +443,4 @@
 \newcommand{\isect}{\cap}
 \newcommand{\union}{\cup}
 \newcommand{\disjunion}{\sqcup}
-\newcommand{\pushout}{\sqcup}
+\newcommand{\pushout}{\sqcup}