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\title{Math 251: Abstract Algebra I \\ In Class Review, Exam \#1}
\date{5 October 2007; exam 8 October 2007}
\maketitle
\begin{problab}{A}
\begin{enumalph}
\item Let $G,H$ be finite groups with $\#G=\#H > 1$. Give an example of a homomorphism $\phi:G \to H$ that is \emph{not} an isomorphism.
\vspace{2in}
\item For every positive even integer $n \in \Z_{>0}$, show that there are at least two nonisomorphic groups of order $n$. Can a group $G$ have $\#G=0$?
\vspace{2in}
\item Exhibit elements $a,b \in D_{2n}$ of order $2$ such that $ab$ has order $n$.
\end{enumalph}
\end{problab}
\newpage
\begin{problab}{B}
Let $G,H$ be groups and $\phi:G \to H$ be a homomorphism.
\begin{enumalph}
\item Prove that the image of $\phi$,
\[ \phi(G)=\{\phi(g) : g \in G\} \]
is a subgroup of $H$.
\vspace{3in}
\item Prove that if $\phi$ is injective, then $G \cong \phi(G)$.
\end{enumalph}
\end{problab}
\end{document}