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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 7.1  \\ Spring 2019 \hfill Due March 6}}}

\large

\vspace{1cm}

\begin{exercise}
Let $G$ be a group and $H < G$.  Prove that if $[G:H]=2$, then $H \lhd G$.
\end{exercise}

\begin{exercise}
Let $G, H$ be groups, $f : G \to H$ a homomorphism.

\begin{itemize}

\item[\bf a.]  If $K \lhd H$, prove that $f^{-1}(K) \lhd G$.

\item[\bf b.]  If $K \lhd G$ and $f$ is surjective, prove that $f(K) \lhd H$.

\end{itemize}

\end{exercise}

\begin{exercise}
Let $G$ be a group, $H < G$ and $N \lhd G$.

\begin{itemize}

\item[\bf a.]  Prove that $HN < G$.

\item[\bf b.]  Prove that $H \cap N \lhd H$.

\item[\bf c.]  Prove that if $H \lhd G$ and $H \cap N$ is trivial, then $hn=nh$ for all $h \in H$ and $n \in N$.

\end{itemize}

\end{exercise}

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