From ec57939416ffdd0537b6182376b27ed6ea48253c Mon Sep 17 00:00:00 2001 From: Mark Date: Sun, 17 Dec 2023 12:10:22 -0800 Subject: [PATCH] Started symmetric group handout --- Advanced/Quotient Groups/main.tex | 8 - Advanced/Symmetric Group/main.tex | 38 +++++ Advanced/Symmetric Group/parts/0 intro.tex | 188 +++++++++++++++++++++ 3 files changed, 226 insertions(+), 8 deletions(-) create mode 100755 Advanced/Symmetric Group/main.tex create mode 100644 Advanced/Symmetric Group/parts/0 intro.tex diff --git a/Advanced/Quotient Groups/main.tex b/Advanced/Quotient Groups/main.tex index 7cebf99..5c8b96a 100755 --- a/Advanced/Quotient Groups/main.tex +++ b/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} diff --git a/Advanced/Symmetric Group/main.tex b/Advanced/Symmetric Group/main.tex new file mode 100755 index 0000000..b9ca834 --- /dev/null +++ b/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} diff --git a/Advanced/Symmetric Group/parts/0 intro.tex b/Advanced/Symmetric Group/parts/0 intro.tex new file mode 100644 index 0000000..9e1c32f --- /dev/null +++ b/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} +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 \ No newline at end of file