Advanced handouts

Add missing file
Co-authored-by: Mark <mark@betalupi.com>
Co-committed-by: Mark <mark@betalupi.com>

src/Advanced/Symmetric Groups/main.tex

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+% use [nosolutions] flag to hide solutions.
+% use [solutions] flag to show solutions.
+\documentclass[
+	solutions,
+	singlenumbering
+]{../../../lib/tex/ormc_handout}
+\usepackage{../../../lib/tex/macros}
+\usetikzlibrary{calc}
+
+\uptitlel{Advanced 2}
+\uptitler{\smallurl{}}
+\title{Symmetric Groups}
+\subtitle{Prepared by Mark on \today{}}
+
+
+
+\def\line#1#2{
+	\draw[line width = 0.3mm, ->, ocyan]
+		(#1)
+		-- ($(#1) + (0, -1)$)
+		-- ($(#2) + (0,1)$)
+		-- (#2);
+}
+
+\begin{document}
+
+	\maketitle
+
+	\input{parts/0 intro}
+	\input{parts/1 cycle}
+	\input{parts/2 groups}
+	\input{parts/3 subgroup}
+
+
+	\section{Bonus problems}
+
+	\problem{}
+	Show that $x \in \mathbb{Z}^+$ has a multiplicative inverse mod $n$ iff $\text{gcd}(x, n) = 1$
+
+	\vfill
+
+	\problem{}
+	Let $\sigma = (\sigma_1 \sigma_2 ... \sigma_k)$ be a $k$-cycle in $S_n$, and let $\tau$ be an arbitrary element of $S_n$. \par
+	Show that $\tau \sigma \tau^{-1}$ = $\bigl(\tau(\sigma_1), \tau(\sigma_2), ..., \tau(\sigma_k)\bigr)$ \par
+	\hint{As usual, $\tau$ is a permutation. Thus, $\tau(x)$ is the value at position $x$ after applying $\tau$.}
+
+	\vfill
+
+	\problem{}
+	Show that the set $\Bigl\{ (1, 2),~ (1,2,...,n) \Bigr\}$ generates $S_n$.
+	\vfill
+
+	% TODO: (a second day?)
+	% alternating group
+	% type and sign and conjugation
+	% isomorphisms & automorphisms
+	% automorphism groups
+\end{document}