Started symmetric group handout

Advanced/Quotient Groups/main.tex

@@ -49,12 +49,4 @@
 	% Q/Z problems (mod generalization)
 	% isomorphism groups (which are iso to symmetric group)
 
-
-	% Another handout:
-	%
-	% symmetric group, number of permutations,
-	% cycle notation, type and sign,
-	% proofs about generators, alternating group
-	% alternating group generators, fun problems.
-
 \end{document}

Advanced/Symmetric Group/main.tex

@@ -0,0 +1,38 @@
+% use [nosolutions] flag to hide solutions.
+% use [solutions] flag to show solutions.
+\documentclass[
+	solutions,
+	singlenumbering
+]{../../resources/ormc_handout}
+\usepackage{../../resources/macros}
+
+
+\uptitlel{Advanced 2}
+\uptitler{Winter 2023}
+\title{Symmetric Groups}
+\subtitle{Prepared by \githref{Mark} on \today{}}
+
+
+
+
+\begin{document}
+
+	\maketitle
+
+	\input{parts/0 intro}
+
+	% cycle notation
+	% decomposition into transpositions
+	% few more problems?
+
+	% inline functions
+	% symmetric group
+	% order & generators
+	% subgroups
+	% alternating group
+
+	% type and sign
+	% isomorphisms & automorphisms
+	% automorphism groups
+
+\end{document}

Advanced/Symmetric Group/parts/0 intro.tex

@@ -0,0 +1,188 @@
+\section{Introduction}
+
+\definition{Intuitive permutations}
+Intuitively, a \textit{permutation} is an ordered arrangement of a set of objects. \par
+For example, $123$, $312$, and $231$ are all permutations of 1, 2, and 3.
+
+\problem{}
+List all permutations on three objects. \par
+How many permutations of $n$ objects are there?
+
+\vfill
+
+
+
+\definition{Formal permutations}<permadef>
+Let $\Omega$ be an arbitrary set of $n$ objects. \par
+A \textit{permutation} on $\Omega$ is a bijective map $f: \Omega \to \Omega$.
+
+\vspace{2mm}
+
+For example, consider the objects 1, 2, and 3. \par
+The permutation $[312]$ is given by a map $f$ defined by the following table:
+\begin{itemize}
+	\item $f(1) = 3$
+	\item $f(2) = 1$
+	\item $f(3) = 2$
+\end{itemize}
+
+Similarly, the \textit{trivial permutation} $[123]$ is given by the identity map $f(x) = x$.
+
+\problem{}
+What map corresponds to the permutation $[321]$?
+
+\vfill
+
+
+
+\problem{}
+Why do we define permutations as a \textit{bijective} map?
+
+
+\vfill
+\pagebreak
+
+
+We can visualize permutations with a diagram we'll call the \say{braid.}
+The arrows in the diagram denote the image of $f$ for each possible input.
+Two examples are below:
+
+\vspace{2mm}
+\hfill
+\begin{tikzpicture}[scale=0.5]
+	\node (1a) at (0, 0.5) {1};
+	\node (2a) at (1, 0.5) {2};
+	\node (3a) at (2, 0.5) {3};
+	\node (4a) at (3, 0.5) {4};
+
+	\node (1b) at (0, -2) {1};
+	\node (3b) at (1, -2) {3};
+	\node (4b) at (2, -2) {4};
+	\node (2b) at (3, -2) {2};
+
+	\draw[line width = 0.3mm, ->, ocyan] (1a.south) -- (1a.south |- 0, -0.5) -- (1b.north |- 0,-1) -- (1b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (2a.south) -- (2a.south |- 0, -0.5) -- (2b.north |- 0,-1) -- (2b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (3a.south) -- (3a.south |- 0, -0.5) -- (3b.north |- 0,-1) -- (3b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (4a.south) -- (4a.south |- 0, -0.5) -- (4b.north |- 0,-1) -- (4b.north);
+\end{tikzpicture}
+\hfill
+\begin{tikzpicture}[scale=0.5]
+	\node (1a) at (0, 0.5) {1};
+	\node (2a) at (1, 0.5) {2};
+	\node (3a) at (2, 0.5) {3};
+	\node (4a) at (3, 0.5) {4};
+
+	\node (2b) at (0, -2) {2};
+	\node (1b) at (1, -2) {1};
+	\node (3b) at (2, -2) {3};
+	\node (4b) at (3, -2) {4};
+
+	\draw[line width = 0.3mm, ->, ocyan] (1a.south) -- (1a.south |- 0, -0.5) -- (1b.north |- 0,-1) -- (1b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (2a.south) -- (2a.south |- 0, -0.5) -- (2b.north |- 0,-1) -- (2b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (3a.south) -- (3a.south |- 0, -0.5) -- (3b.north |- 0,-1) -- (3b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (4a.south) -- (4a.south |- 0, -0.5) -- (4b.north |- 0,-1) -- (4b.north);
+\end{tikzpicture}
+\hfill\null
+
+Note that in all our examples thus far, the objects in our set have an implicit order.
+This is only for convenience. The elements of $\Omega$ are not ordered (it is a \textit{set}, after all),
+and we may present them however we wish.
+
+
+
+
+
+
+
+\vspace{1cm}
+
+For example, consider the diagrams below. \par
+On the left, 1234 are ordered as usual. In the middle, they are ordered alphabetically. \par
+The rightmost diagram uses arbitrary, meaningless labels.
+
+\vspace{2mm}
+\hfill
+\begin{tikzpicture}[scale=0.5]
+	\node (1a) at (0, 0.5) {1};
+	\node (2a) at (1, 0.5) {2};
+	\node (3a) at (2, 0.5) {3};
+	\node (4a) at (3, 0.5) {4};
+
+	\node (2b) at (0, -2) {2};
+	\node (1b) at (1, -2) {1};
+	\node (3b) at (2, -2) {3};
+	\node (4b) at (3, -2) {4};
+
+	\draw[line width = 0.3mm, ->, ocyan] (1a.south) -- (1a.south |- 0, -0.5) -- (1b.north |- 0,-1) -- (1b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (2a.south) -- (2a.south |- 0, -0.5) -- (2b.north |- 0,-1) -- (2b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (3a.south) -- (3a.south |- 0, -0.5) -- (3b.north |- 0,-1) -- (3b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (4a.south) -- (4a.south |- 0, -0.5) -- (4b.north |- 0,-1) -- (4b.north);
+\end{tikzpicture}
+\hfill
+\begin{tikzpicture}[scale=0.5]
+	\node (4a) at (0, 0.5) {4};
+	\node (1a) at (1, 0.5) {1};
+	\node (3a) at (2, 0.5) {3};
+	\node (2a) at (3, 0.5) {2};
+
+	\node (1b) at (0, -2) {1};
+	\node (4b) at (1, -2) {4};
+	\node (3b) at (2, -2) {3};
+	\node (2b) at (3, -2) {2};
+
+	\draw[line width = 0.3mm, ->, ocyan] (1a.south) -- (1a.south |- 0, -0.5) -- (1b.north |- 0,-1) -- (1b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (2a.south) -- (2a.south |- 0, -0.5) -- (2b.north |- 0,-1) -- (2b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (3a.south) -- (3a.south |- 0, -0.5) -- (3b.north |- 0,-1) -- (3b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (4a.south) -- (4a.south |- 0, -0.5) -- (4b.north |- 0,-1) -- (4b.north);
+\end{tikzpicture}
+\hfill
+\begin{tikzpicture}[scale=0.5]
+	\node (1a) at (0, 0.5) {$\triangle$};
+	\node (2a) at (1, 0.5) {$\divideontimes$};
+	\node (3a) at (2, 0.5) {$\circledcirc$};
+	\node (4a) at (3, 0.5) {$\boxdot$};
+
+	\node (2b) at (0, -2) {$\divideontimes$};
+	\node (1b) at (1, -2) {$\triangle$};
+	\node (3b) at (2, -2) {$\circledcirc$};
+	\node (4b) at (3, -2) {$\boxdot$};
+
+	\draw[line width = 0.3mm, ->, ocyan] (1a.south) -- (1a.south |- 0, -0.5) -- (1b.north |- 0,-1) -- (1b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (2a.south) -- (2a.south |- 0, -0.5) -- (2b.north |- 0,-1) -- (2b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (3a.south) -- (3a.south |- 0, -0.5) -- (3b.north |- 0,-1) -- (3b.north);
+	\draw[line width = 0.3mm, ->, ocyan] (4a.south) -- (4a.south |- 0, -0.5) -- (4b.north |- 0,-1) -- (4b.north);
+\end{tikzpicture}
+\hfill\null
+
+It shouldn't be hard to see that despite the different \say{output} order (2134 and 1432), \par
+the same permutation is depicted in all three diagrams. This example demonstrates two things:
+\begin{itemize}[itemsep=2mm]
+	\item First, the items of our set do not have any meaning. \par
+	$\Omega$ is just a set of arbitrary \textit{things}, which we may label however we like.
+
+	\item Second, permutations are verbs. We do not care about the \say{output} of a certain permutation,
+	we care about what it \textit{does}. We could, for example, describe the permutation above as
+	\say{swap the first two of four elements.}
+\end{itemize}
+
+\vspace{2mm}
+
+\vspace{1cm}
+
+Why, then, do we order our elements when we talk about permutations? As noted before, this is for convenience.
+If we assign a natural order to the elements of $\Omega$ (say, 1234), we can identify permutations by simply listing
+their output:
+Clearly, $[1234]$ represents the trivial permutation, $[2134]$ represents \say{swap first two,}
+and $[4123]$ represents \say{cycle left.}
+
+\problem{}
+Draw braids for $[4123]$ and $[2341]$.
+
+\vfill
+
+
+Finally, note that permutations (as defined in \ref{permadef}) are \textit{not} \say{orderings of a certain set.} \par
+They are defined as \textit{bijective maps}, which can be \textit{thought of} as orderings. \par
+Remember: permutations are verbs!
+
+\pagebreak