Finished symmetric group handout

Advanced/Symmetric Group/main.tex

@@ -28,19 +28,31 @@
 
 	\input{parts/0 intro}
 	\input{parts/1 cycle}
+	\input{parts/2 groups}
+	\input{parts/3 subgroup}
 
 
-	% decomposition into transpositions
-	% few more problems?
+	\section{Bonus problems}
 
-	% inline functions
-	% symmetric group
-	% order & generators
-	% subgroups
+	\problem{}
+	Show that $x$ has a multiplicative inverse mod $n$ iff $\text{gcd}(x, n) = 1$
+
+	\vfill
+
+	\problem{}
+	Let $\sigma = (\sigma_1 \sigma_2 ... \sigma_k)$ be a $k$-cycle in $S_n$, and let $\tau$ be an arbitrary element of $S_n$. \par
+	Show that $\tau \sigma \tau^{-1}$ = $\bigl(\tau(\sigma_1), \tau(\sigma_2), ..., \tau(\sigma_k)\bigr)$ \par
+	\hint{As usual, $\sigma$ is a permutation. Thus, $\sigma(x)$ is the value at position $x$ after applying $\sigma$.}
+
+	\vfill
+
+	\problem{}
+	Show that the set $\Bigl\{ (1, 2),~ (1,2,...,n) \Bigr\}$ generates $S_n$.
+	\vfill
+
+	% TODO: (a second day?)
 	% alternating group
-
-	% type and sign
+	% type and sign and conjugation
 	% isomorphisms & automorphisms
 	% automorphism groups
-
 \end{document}