Symmetric edits

Advanced/Symmetric Group/parts/2 groups.tex

@@ -6,11 +6,11 @@ Before we continue, we must introduce a bit of notation:
 	\item $S_n$ is the set of permutations on $n$ objects.
 	\item $\mathbb{Z}_n$ is the set of integers mod $n$.
 
-	\item $\mathbb{Z}_n^\times$ is the set of integers mod $n$ with multiplicative inverses, which is \par
-	the set of integers smaller than $n$ and coprime to $n$\footnotemark{}\hspace{-1ex}. \par
+	\item $\mathbb{Z}_n^\times$ is the set of integers mod $n$ with multiplicative inverses. \par
+	In other words, it is the set of integers smaller than $n$ and coprime to $n$.\footnotemark{} \par
 	For example, $\mathbb{Z}_{12}^\times = \{1, 5, 7, 11\}$.
 
-	\footnotetext{We proved this in another handout, but you make take it as fact here.}
+	\footnotetext{We proved this in another handout, but you may take it as fact here.}
 \end{itemize}
 
 \problem{}
@@ -26,7 +26,7 @@ Groups always have the following properties:
 
 \begin{enumerate}
 	\item $G$ is closed under $\ast$. In other words, $a, b \in G \implies a \ast b \in G$.
-	\item $\ast$ is associative: $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a,b,c \in G$
+	\item $\ast$ is \textit{associative}: $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a,b,c \in G$
 	\item There is an \textit{identity} $e \in G$, so that $a \ast e = a \ast e = a$ for all $a \in G$.
 	\item For any $a \in G$, there exists a $b \in G$ so that $a \ast b = b \ast a = e$. $b$ is called the \textit{inverse} of $a$. \par
 	This element is written as $-a$ if our operator is addition and $a^{-1}$ otherwise.
@@ -42,7 +42,7 @@ Is $(\mathbb{Z}_5, -)$ a group? \par
 
 
 \problem{}
-What is the smallest group?
+What is the group with the fewest elements?
 
 \begin{solution}
 	Let $(G, \star)$ be our group, where $G = \{x\}$ and $\star$ is defined by $x \star x = x$
@@ -63,7 +63,10 @@ What is the smallest group?
 
 
 
+\problem{}
+Show that function composition is associative
 
+\vfill
 
 
 \problem{}
@@ -82,7 +85,7 @@ The smallest such $n$ defines the \textit{order} of $g$.
 
 \begin{examplesolution}
 	We've already done a special case of this problem! \par
-	Look back through the handout and find it, then rewrite your proof for an arbitrary group.
+	Find it in this handout, then rewrite your proof for an arbitrary (finite) group.
 \end{examplesolution}
 
 
@@ -116,34 +119,25 @@ We say $g$ is a \textit{generator} if every other element of $G$ may be written
 Say the size of a group $G$ is $n$. \par
 If $g$ is a generator, what is its order? \par
 Provide a proof.
+\vfill
+
+
+
+\problem{}
+Find the two generators in $(\mathbb{Z}, +)$ \par
+Then, find all generators of $(\mathbb{Z}_5, +)$
+\vfill
+
+
+\problem{}
+How many groups have only one generator?
 
 \begin{solution}
-	The order of a generator must equal the order of its group.
+	Only one: the trivial group. The inverse of a generator is also a generator!
 \end{solution}
 
 \vfill
 
-\problem{}
-Find the only generator of $(\mathbb{Z}^+, +)$ \par
-Then, find all generators of $(\mathbb{Z}_5, +)$
-
-\vfill
-\pagebreak
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
 
 \definition{}
 Let $S$ be a subset of the elements in $G$. \par
@@ -168,13 +162,4 @@ We've already found a few generating sets of $S_n$. What are they?
 \end{solution}
 
 \vfill
-
-\problem{}
-Find the smallest set that generates $(\mathbb{Z}^+, +)$. \par
-\vfill
-
-\problem{}
-Find the smallest set that generates $(\mathbb{Z}, +)$. \par
-\vfill
-
 \pagebreak