Typo fixes

Advanced/Symmetric Group/parts/1 cycle.tex

@@ -57,14 +57,14 @@ In other words, show that there is always an integer $n$ so that $f^n(x) = x$.
 
 \vfill
 
-\definition{Composition}
+\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 $fg$.
 
 \problem{}
 What is $[1324][4321]$? \par
 How about $[321][213][231]$? \par
-\hint{composition is left-associative, so we evaluate $abc$ as $(ab)c$}
+\hint{is composition is left or right-associative? See \ref{compdef}}
 
 
 \vfill
@@ -440,7 +440,7 @@ How about $(123)$? And $(4231)$? \par
 
 
 \problem{}
-Say $\sigma$ is a permutation composed of cycles $\sigma_1\sigma_2...\sigma_k$. \par
+Say $\sigma$ is a permutation composed of disjoint cycles $\sigma_1\sigma_2...\sigma_k$. \par
 Say we know the order of all $\sigma_i$. What is the order of $\sigma$?
 
 \begin{solution}
@@ -479,7 +479,8 @@ Show that any permutation is a product of transpositions.
 Show that any permutation is a product of transpositions of the form $(1, k)$. \par
 
 \begin{solution}
-	Use \ref{simpletrans} and rewrite each $(a, b)$ as $(1, a)(1, b)(1, a)$.
+	Use \ref{simpletrans} and rewrite each $(a, b)$ as $(1, a)(1, b)(1, a)$. \par
+	Showing that $(a, b) = (1, a)(1, b)(1, a)$ is fairly easy.
 \end{solution}
 
 \vfill
@@ -491,7 +492,8 @@ Show that any permutation is a product of transpositions of the form $(1, k)$. \
 Show that any transposition $(a, b)$ is equal to the product $(a, a+1)(a+1, b)(a, a+1)$.
 
 \begin{solution}
-	TODO
+	This is the same as the $(1, a)(1, b)(1, a)$ case above, but we use $a + 1$
+	as a \say{working slot} instead of $1$.
 \end{solution}
 
 \vfill