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\section*{Math 3140: Homework 10}

\begin{enumerate}
\item[A.] \hfill

\begin{enumerate}
\item[15.10.] Let $H\subseteq G$ be a subgroup, and suppose $H=\langle X\rangle$ for some subset $X\subseteq H$ and $G=\langle Y\rangle$ for some subset $Y\subseteq G$.  Show that $H\triangleleft G$ if $yxy^{-1}\in H$ for all $x\in X$ and $y\in Y$.
\item[15.14.] \begin{enumerate}
\item Show that every element in $\QQ/\ZZ$ has finite order.
\item Show that the only element in $\RR/\QQ$  that has finite order is the identity.
\end{enumerate}
\end{enumerate}


\item[B.] \hfill

\begin{enumerate}
\item[16.4.]  Let $A\triangleleft G$ and $B\triangleleft H$.  Show that $(A\times B)\triangleleft (G\times H)$ and 
$$(G\times H)/(A\times B)\cong (G/A)\times (H/B).$$
(see how heavily you can use the isomorphism theorems to prove this).
\item[16.6.] 
\item[16.11.]  (Fourth Isomorphism Theorem) Let $\varphi:G\rightarrow H$ be a surjective homomorphism with kernel $K$.  
\begin{enumerate}
\item[(a)] For every subgroup $B\subseteq H$, show that the set
$$\varphi^{-1}(B)=\{g\in G\ \mid\ \varphi(g)\in B\}$$
is a subgroup of $G$ that contains $K$.
\item[(b)] Show that there is a bijection between
$$\left\{\begin{array}{c} \text{Subgroups $A\subseteq G$}\\ \text{such that $K\subseteq A$}\end{array}\right\}\longleftrightarrow \left\{\begin{array}{c} \text{Subgroups $B\subseteq H$}\end{array}\right\}.$$
\end{enumerate}
\item[16.12] An \emph{maximal} normal subgroup $N$ of $G$ is a normal subgroup such that if $H\supseteq N$ is a normal subgroup of $G$, then $H=N$ or $H=G$.  Show that $N$ is a maximal normal subgroup of $G$ if and only if $G/N$ is simple.
\end{enumerate}
\end{enumerate}
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