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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 8.1  \\ Spring 2019 \hfill Due March 20}}}

\large

\vspace{1cm}

\begin{exercise}
Let $G_1$ and $G_2$ be groups, with $H_1 \lhd G_1$ and $H_2 \lhd G_2$.  Use the First Isomorphism Theorem to prove that $H_1 \times H_2 \lhd G_1 \times G_2$ and 
\[
(G_1 \times G_2)/(H_1 \times H_2) \cong (G_1/H_1) \times (G_2/H_2).
\]  
[{\em Suggestion:}  Use the canonical epimorphisms $G_i \to G_i/H_i$ to construct a homomorphism $G_1 \times G_2 \to (G_1/H_1) \times (G_2/H_2)$, and compute its kernel.]
\end{exercise}

\begin{exercise}
Let $m, n \in \N$ be relatively prime.  Use the First Isomorphism Theorem to prove the Chinese Remainder Theorem:
\[
\Z / mn \Z \cong (\Z /m \Z) \times (\Z / n \Z).
\]
[{\em Suggestion:}  Use the canonical epimorphisms from $\Z$ onto $\Z /m \Z$ and $\Z / n \Z$ to construct a homomorphism $\Z \to (\Z /m \Z) \times (\Z / n \Z)$, and compute its kernel.  Use the Pigeonhole Principle to prove that the induced monomorphism is surjective.]
\end{exercise}

\begin{exercise}
Let $G$ be a group, $H < G$ and $N \lhd G$.  Prove the Second Isomorphism Theorem:
\[
H/(H \cap N) \cong HN/N.
\]
[{\em Suggestion:}  Compose the inclusion $H \hookrightarrow HN$ with the canonical surjection $HN \to HN/N$, and compute the kernel.  You'll need a separate argument to show that the induced monomorphism is surjective.]
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