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2024-01-03 22:25:15 -08:00
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Advanced/Symmetric Group/parts/1 cycle.tex
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@ -2,8 +2,10 @@
\section{Cycle Notation}
\definition{Order}
The \textit{order} of a permutation $f$ is the smallest $n$ so that $f^n(x) = x$ for all $x$. \par
In other words, if we repeat this permutation $n$ times, we get back to where we started.
The \textit{order} of a permutation $f$ is the smallest positive $n$ so that $f^n(x) = x$ for all $x$. \par
In other words: if we repeat this permutation $n$ times, we get back to where we started. \par
Note that the order is given by the \textit{smallest} positive integer $n$. There may be more than one!
\vspace{2mm}
@ -44,7 +46,7 @@ Naturally, the identity permutation has order one.
\problem{}
What is the order of $[2314]$? \par
How about $[4321]$? \par
\note[Note]{You shouldn't need to draw any braids to solve this problem.}
\note[Note]{You shouldn't need to draw any strings to solve this problem.}
\vfill
@ -168,6 +170,9 @@ The permutation $[431265]$ is a bit more interesting---it contains of two cycles
\end{center}
Another name we'll often use for two-cycles is \textit{transposition}. \par
Any permutation that swaps two adjacent elements is called a transposition. \par
\problem{}
Find all cycles in $[5342761]$.
@ -417,7 +422,8 @@ Be careful.
\problem{}
Look at the last two permutations in \ref{insquare}, $(1234)$ and $(3412)$. \par
These are \textit{identical}---they are the same cycle written in two different ways. \par
List all other ways to write this cycle. \hint{There are two more.}
List all other ways to write this cycle. \hint{There are two more.} \par
\note{Also, note that the last two permutations in \ref{insquare} are the same.}
\pagebreak