Finished symmetric group handout
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@ -1,7 +1,7 @@
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\section{Cycle Notation}
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\definition{}
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\definition{Order}
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The \textit{order} of a permutation $f$ is the smallest $n$ so that $f^n(x) = x$ for all $x$. \par
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In other words, if we repeat this permutation $n$ times, we get back to where we started.
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@ -36,8 +36,9 @@ For example, consider $[2134]$. This permutation has order $2$, as we clearly se
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\line{4b}{4c}
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\end{tikzpicture}
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\end{center}
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Of course, swapping the first two elements twice results in the identity map. \par
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$[2134]$ happens to be its own inverse. Also, the order of $[1234]$ is (of course) one.
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Of course, swapping the first two elements of a list twice changes nothing. \par
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Thus, $[2134]$ is its own inverse, and has an order of two. \par
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Naturally, the identity permutation has order one.
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\problem{}
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@ -52,6 +53,18 @@ How about $[4321]$? \par
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Show that all permutations (on a finite set) have a well-defined order. \par
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In other words, show that there is always an integer $n$ so that $f^n(x) = x$.
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\vfill
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\definition{Composition}
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The \textit{composition} of two permutations $f$ and $g$ is the permutation $h(x) = f(g(x))$. \par
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The usual notation for this is $f \circ g$, but we'll simply write $fg$.
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\problem{}
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What is $[1324][4321]$? \par
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How about $[321][213][231]$? \par
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\hint{composition is left-associative, so we evaluate $abc$ as $(ab)c$}
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\vfill
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\pagebreak
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@ -74,7 +87,7 @@ We need something better.
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\definition{}
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\definition{Cycles}
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Any permutation is composed of a number of \textit{cycles}. \par
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For example, consider the permutation $[2134]$, which consists of one two-cycle: $1 \to 2 \to 1$ \par
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@ -217,7 +230,7 @@ Find all cycles in $[5342761]$.
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\problem{}
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What permutation (on five objects) consists of the cycles $3 \to 5 \to 3$ and $1 \to 2 \to 4 \to 1$?
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What permutation (on five objects) is formed by the cycles $3 \to 5 \to 3$ and $1 \to 2 \to 4 \to 1$?
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\begin{solution}
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@ -271,7 +284,7 @@ What permutation (on five objects) consists of the cycles $3 \to 5 \to 3$ and $1
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\vfill
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\pagebreak
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\definition{}
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\definition{Cycle Notation}
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We now have a solution to our problem of notation.
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Instead of referring to permutations using their output, we will refer to them using their \textit{cycles}.
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@ -355,12 +368,20 @@ applying the permutation $[431265]$ is the same as applying $(1324)$, then apply
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\end{tikzpicture}
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\end{center}
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Any permutation $\sigma$ may be written as a product (i.e, composition) of disjoint cycles $\sigma_1\sigma_2...\sigma_k$. \par
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Make sure you believe this fact. If you don't, ask an instructor. \par
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Also, the identity $f(x) = x$ is written as $()$ in cycle notation.
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\problem{}
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Convince yourself that disjoint cycles commute. \par
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That is, $(1324)(56) = (56)(1324)$ since $(1324)$ and $(56)$ do not overlap.
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That is, that $(1324)(56) = (56)(1324) = [431265]$ since $(1324)$ and $(56)$ do not overlap. \par
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\problem{}
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\problem{}<insquare>
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Write the following in square-bracket notation.
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\begin{itemize}
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\item $(12)$ \tab~\tab on a set of 2 elements
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@ -372,9 +393,15 @@ Write the following in square-bracket notation.
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\item $(1234)$ \tab on a set of 4 elements
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\item $(3412)$ \tab on a set of 4 elements
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\end{itemize}
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\note{
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Note that $(12)$ refers the \say{swap first two} permutation on a set of \textit{any} size. \\
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We can now use the same name for the same permutation on two different sets! \\
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}
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\vfill
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\problem{}
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Write the following in square-bracket notation.
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Be careful.
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@ -383,5 +410,117 @@ Be careful.
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\item $(243)(13)$ \tab on a set of 4 elements
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\end{itemize}
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\vfill
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\problem{}
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Look at the last two permutations in \ref{insquare}, $(1234)$ and $(3412)$. \par
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These are \textit{identical}---they are the same cycle written in two different ways. \par
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List all other ways to write this cycle. \hint{There are two more.}
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\pagebreak
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\problem{}
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What is the inverse of $(12)$? \par
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How about $(123)$? And $(4231)$? \par
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\note{
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Note that again, we don't need to know how big our set is. \\
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The inverse of $(12)$ is the same in all sets.
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}
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\vfill
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\problem{}
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Say $\sigma$ is a permutation composed of cycles $\sigma_1\sigma_2...\sigma_k$. \par
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Say we know the order of all $\sigma_i$. What is the order of $\sigma$?
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\begin{solution}
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$\text{lcm}\Bigl(\text{ord}(\sigma_1),~ \text{ord}(\sigma_2),~ ..., ~ \text{ord}(\sigma_k)\Bigr)$
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\end{solution}
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\vfill
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\problem{}<cycletrans>
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Show that any cycle $(123...n)$ is equal to the product $(12)(23)...(n-1, n)$.
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\begin{solution}
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TODO
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\end{solution}
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\vfill
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\problem{}
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Write $(7126453)$ as a product of transpositions. \par
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\vfill
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\pagebreak
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\problem{}<simpletrans>
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Show that any permutation is a product of transpositions.
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\begin{solution}
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Use \ref{cycletrans}.
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\end{solution}
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\vfill
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\problem{}
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Show that any permutation is a product of transpositions of the form $(1, k)$. \par
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\begin{solution}
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Use \ref{simpletrans} and rewrite each $(a, b)$ as $(1, a)(1, b)(1, a)$.
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\end{solution}
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\vfill
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\pagebreak
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\problem{}
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Show that any transposition $(a, b)$ is equal to the product $(a, a+1)(a+1, b)(a, a+1)$.
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\begin{solution}
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TODO
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\end{solution}
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\vfill
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\problem{}
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Show that any permutation is a product of adjacent transpositions. \par
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(An \textit{adjacent transposition} swaps two adjacent elements, and thus looks like $(n, n+1)$)
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\begin{solution}
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As before, we will use \ref{simpletrans} and rewrite the transpositions it produces in a form that fits the problem.
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We thus need to show that every transposition $(a, b)$ is a product of adjacent transpositions.
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\vspace{8mm}
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In the proof below, assume that $a < b$ and perform induction on $b - a$. \par
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\textbf{Base Case:}\par
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If $b - a = 1$, we clearly see that $(a, b)$ is a product of adjacent. \par
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In fact, it \textit{is} an adjacent transposition.
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\vspace{4mm}
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\textbf{Induction:}\par
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Now, say $b - a = n + 1$. \par
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Assume that all $(a, b)$ where $b - a \leq n$ are products of adjacent transpositions.\par
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Note that $(a, b) = (a, a+1)(a+1, b)(a, a+1)$.
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\vspace{2mm}
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$(a, a+1)$ is an adjacent transposition, and $b - (a+1) = n$. \par
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Thus, $(a, b)$ is a product of adjacent transpositions.
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\end{solution}
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\vfill
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\pagebreak
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