Symmetric group edits

Advanced/Symmetric Group/parts/1 cycle.tex

@@ -2,10 +2,8 @@
 \section{Cycle Notation}
 
 \definition{Order}
-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!
-
+The \textit{order} of a permutation $f$ is the \textbf{smallest} positive $n$ so that $f^n(x) = x$ for all $x$. \par
+If we repeatedly apply a permutation with order $n$, we will get back to where we started after $n$ steps. \par
 
 \vspace{2mm}
 
@@ -38,9 +36,8 @@ For example, consider $[2134]$. This permutation has order $2$, as we clearly se
 	\line{4b}{4c}
 \end{tikzpicture}
 \end{center}
-Of course, swapping the first two elements of a list twice changes nothing. \par
-Thus, $[2134]$ is its own inverse, and has an order of two. \par
-Naturally, the identity permutation has order one.
+Swapping the first two elements of a list twice changes nothing. \par
+Thus, $[2134]$ has an order of two.
 
 
 \problem{}
@@ -51,7 +48,7 @@ How about $[4321]$? \par
 
 \vfill
 
-\problem{Bonus}
+\problem{}
 Show that all permutations (on a finite set) have a well-defined order. \par
 In other words, show that there is always an integer $n$ so that $f^n(x) = x$.
 
@@ -59,12 +56,17 @@ In other words, show that there is always an integer $n$ so that $f^n(x) = x$.
 
 \definition{Composition}<compdef>
 The \textit{composition} of two permutations $f$ and $g$ is the permutation $h(x) = f(g(x))$. \par
-The usual notation for this is $f \circ g$, but we'll simply write $fg$.
+We'll denote this as $fg$---that is, by simply writing the permutations we're composing next to each other.
+
+\problem{}
+Show that function composition is associative. \par
+That is, show that $f(gh) = (fg)h$.
+
+\vfill
 
 \problem{}
 What is $[1324][4321]$? \par
 How about $[321][213][231]$? \par
-\hint{is composition is left or right-associative? See \ref{compdef}}
 
 
 \vfill