% use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions.
\documentclass[
	solutions,
	singlenumbering
]{../../resources/ormc_handout}
\usepackage{../../resources/macros}
\usetikzlibrary{calc}

\uptitlel{Advanced 2}
\uptitler{Winter 2023}
\title{Symmetric Groups}
\subtitle{Prepared by \githref{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}
