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\begin{document}
\title{Homework $\#$4}
\author{Due 10/16/13 at start of class }
\maketitle
\subsection*{Problem 1} Let $G$ be a group and let $H$ be a subgroup of $G$.
\begin{itemize}
\item[(a)] Let $\phi:G\rightarrow H$ be a homomorphism. Prove that $\ker(\phi)\trianglelefteq G$.
\item[(b)] Prove that $N_G(H)\leq G$ and $H\leq N_G(H)$. 
\end{itemize}  

\subsection*{Problem 2 (D$\&$F 3.1.5)} Prove that the order of the element $gH$ in the quotient group $G/H$ is $n$, for $n$ the smallest positive integer such that $g^n\in H$ (and $gH$ has infinite order if no such positive integer exists). Give an example to show that the order of $gH$ in $G/H$ may be strictly smaller than the order of $g$ in $G$.

\noindent [Hint: you may use without proof that $(gH)^a=g^a H$ for all $a\in \mathbb{Z}$.]

 
\subsection*{Problem 3 (D$\&$F 3.1.22)} Let $G$ be a group.
\begin{itemize}
\item[(a)] Prove that if $H$ and $K$ are normal subgroups of $G$ then their intersection is also a normal subgroup of $G$.
\item[(b)] Prove that the intersection of an arbitrary nonempty collection of normal subgroups of a group is a normal subgroup (do not assume the collection is countable).
\end{itemize}

\subsection*{Problem 4 (D$\&$F 3.1.42)} Let $H$ and $K$ be normal subgroups of $G$ with $H\cap K=1$. Prove that $xy=yx$ for all $x\in H$ and $y\in K$.

\noindent [Hint: show $x^{-1}y^{-1}xy\in H\cap K$.] 

\subsection*{Problem 5 (D$\&$F 3.2.11)} Let $H\leq K\leq G$. Prove that $[G:H]=[G:K][K:H]$ (do not assume $G$ is finite).

\subsection*{Problem 6 D$\&$F 3.2.16} Use Lagrange's theorem in the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^\times$ to prove \textit{Fermat's Little Theorem}: if $p$ is prime then $a^p\equiv a(\Mod p)$ for all $a\in \mathbb{Z}$.

\subsection*{Problem 7 (D$\&$F 3.2.19)} Prove that if $N$ is a normal subgroup of the finite group $G$ and $(\vert N\vert ,[G:N])=1$ then $N$ is the unique subgroup of $G$ of order $\vert N\vert$. 

\subsection*{Problem 8 (D$\&$F 3.1.36)} Prove that if $G/Z(G)$ is cyclic then $G$ is abelian.

\subsection*{Challenge Problem} Let $G$ be a group of order $pq$. Prove that either $G$ is abelian or $Z(G)=1$.
\end{document}