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\begin{document}
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\begin{center}\textbf{Math 31 Homework 5}\\Due August 4, 2018\end{center}

\begin{enumerate}

\item Chapter 15, exercise C5.

\item Chapter 15, exercise D1.

\item Chapter 14, exercise C4.

\item Let $G,H$ be groups. Prove that $(G\times H)/(G\times\{e\})\cong H$.

\item Let $G$ be an abelian group of order $120$ that has exactly three elements of order $2$.
\begin{enumerate} 
\item List all possible abelian groups (up to isomorphism) of order $120$.
\item Determine the isomorphism class of $G$.
\end{enumerate}

\item Prove the last part of the Third Isomorphism Theorem: If $K\unlhd G$ and $H\unlhd G$ such that $K\subseteq H\subseteq G$, then $$(G/K)/(H/K)\cong G/H.$$ You may use the previous parts of the Third Isomorphism Theorem. \\(\textit{Hint}: Find a homomorphism between related groups, then use the First Isomorphism Theorem to get the result.)

\end{enumerate}
\end{document}