59 lines
1.3 KiB
TeX
Executable File
59 lines
1.3 KiB
TeX
Executable File
% 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{\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}
|