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\centerline{Mathematics 243, section 3 -- Algebraic Structures}
\centerline{Problem Set 9}
\centerline{{\bf due:}  Friday, November 30}
\vskip 10pt
\noindent
{\it `A' Section}
\bigskip
\begin{enumerate}
\item[1.]  For each of the following values of $n$, 
\begin{itemize}
\item Find all distinct generators of the group $(\mathbb{Z}_n,+)$, 
\item Find all subgroups of $(\mathbb{Z}_n,+)$ and their orders
\item Find all elements of $(\mathbb{Z}_n^\times,\cdot)$ and their
orders (for the \emph{multiplication operation} mod $n$ now) 
\end{itemize}
$$n = 13, 16, 30$$
(Use the ``big theorem'' on cyclic groups for as much of this as possible.
It is not necessary to do a lot of computations in most cases.)
\item[2.]  Let $\varphi : \mathbb{Z}_{18} \to \mathbb{Z}_{9}$
be defined by $\varphi([x]) = [3x]$.  
\begin{enumerate}
\item[(a)]  Verify that $\varphi$ is a group homomorphism.
\item[(b)]  Determine the kernel of $\varphi$.
\item[(c)]  Determine the image of $\varphi$.
\end{enumerate}
\end{enumerate}
\vskip 10pt
\noindent
{\it `B' Section}
\vskip 10pt
\begin{enumerate}
\item[1.]  Let $G$ be a group and consider the mapping $\varphi : G \to G$
defined by $\varphi(x) = x^{-1}$.  Show that $\varphi$ is always one-to-one and
onto, but that $\varphi$ is an isomorphism of groups if and only if $G$ is 
an \emph{abelian} group.
\item[2.]  An \emph{automorphism} of a group $G$ is an isomorphism of 
groups $\varphi : G \to G$ (that is, the domain and the range are both the 
same group $G$).
\begin{enumerate}  
\item[(a)]  Let $A = \{a,b,c\}$ and $G = \mathcal{S}(A)$ be the 
group of permutations of $A$.  Show that $\varphi : G \to G$ defined
by $\varphi(f) = R_a \circ f \circ R_a$ is an automorphism of $G$.  
\item[(b)]  Show that the collection of all automorphisms of a general group $G$ is 
itself a group under the operation of function composition.
\item[(c)]  Show if $G$ is a general group and $g \in G$, then the 
conjugation mapping defined by $\varphi_g(x) = gxg^{-1}$ is an automorphism of $G$. (Note
that the example in part (a) has this form.)
\item[(d)]  Show that the collection of $\varphi_g$ for all $g \in G$ (as in part (c))
is a \emph{subgroup} of the group of automorphisms of $G$.
\end{enumerate}
\item[3.]  We can consider isomorphism of groups as a relation on 
the collection of all groups:  $G R H \Leftrightarrow $ there exists an 
isomorphism $\varphi : G \to H$.  Show that isomorphism of groups is an 
\emph{equivalence relation} on the collection of all groups.  
\item[4.]  Let $G = \langle a \rangle$ be a cyclic group and let 
$\varphi : G \to H$ be a group homomorphism.  Show that if we know the one element
$\varphi(a)$, then we know where $\varphi$ maps every element of $G$.  
\end{enumerate}
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