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\section*{Math 403/503 Spring 2024}
\subsection*{Homework 2, due January 31}
\begin{enumerate}
\item Consider the abelian group $G=\mathbb{Z}_{12}$. Let $N=\{\bar 0,\bar 3,\bar 6,\bar 9\}$ (where $\bar a=[a]_{12}$ denotes the congruence class of $a$ modulo $12$). Find the Cayley table for the group $G/N$. Compare it to the Cayley table for $\mathbb{Z}_3$.
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\item Let $G=D_n=\{1,r,r^2,\ldots,r^{n-1}, s,sr,sr^2,\ldots,sr^{n-1}\}$ be the dihedral group of order $2n$. Let $N=\langle r\rangle=\{1,r,r^2,\ldots,r^{n-1}\}$. Show that $N$ is a normal subgroup of $G$ and that $G/N\cong \mathbb{Z}_2$.
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\item Let $G_1$ and $G_2$ be groups with identity elements denoted $e_1$ and $e_2$ respectively. Let $G=G_1\times G_2$. 
\begin{enumerate}[{\rm (a)}]
\item Show that $\widetilde{G}_1=\{(x,e_2)\mid x\in G_1\}$ is a normal subgroup of $G$. 
\item Show that $G/\widetilde{G}_1$ is isomorphic to the group $G_2$. \emph{Hint:} Find a surjective homomorphism $\varphi:G\to G_2$ with kernel equal to $\widetilde{G}_1$. Then use the First Isomorphism Theorem.
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\item Show that $\mathbb{Z}_{12}\cong \mathbb{Z}_3\times \mathbb{Z}_4$. \emph{Hint:} Consider the homomorphism $\varphi:\mathbb{Z}\to\mathbb{Z}_3\times\mathbb{Z}_4$, $\varphi(a)=\big([a]_3,[a]_4\big)$.
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\item Let $M$ be a normal subgroup of a group $G$ and let $N$ be a normal subgroup of a group $H$. If $\varphi:G\to H$ is a homomorphism such that $\varphi(M)\subseteq N$, prove that the map $\psi:G/M\to H/N$ given by $\psi(gM)=\varphi(g)N$ is a well-defined homomorphism.
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