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\title{Math 420-620\\Fall 2012\\Homework 4}
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\centerline{\it Due Monday September 24, 2012.}

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\noindent 1. Let $G$ be a group and $x,y\in G$. We say that the {\it commutator} of $x$ and $y$ (sometimes denoted $[x,y]$) is $[x,y]=x^{-1}y^{-1}xy$. Globally, we define the {\it commutator subgroup} of $G$ to be

\[
G^\prime=\langle [x,y]\vert x,y\in G\rangle.
\]

\begin{itemize}
\item[a)] (5 pt) Show that the elements $x,y\in G$ commute if and only if $[x,y]=1$.
\item[b)] (5 pt) Show that $G^\prime$ is a normal subgroup of $G$.
\item[c)] (5 pt) Show that if $N$ is a normal subroup of $G$, then $G/N$ is abelian if and only if $N$ contains $G^\prime$.
\end{itemize}

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\noindent 2. (5 pt) Let $G$ be a group and $\phi:G\longrightarrow H$ be a homomorphism. Show that $\text{Z}(G)$ and $\text{ker}(\phi)$ are normal subgroups of $G$.

\centerline{}

\noindent 3. (5 pt) Let $G$ be a group and $H$ a subgroup of $G$ such that $[G:H]=2$. Show that $H\unlhd G$. Is the same true if $[G:H]=3$?

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\noindent 4. (5 pt) Let $G$ be a group with center $\text{Z}(G)$. Show that $G$ is abelian if and only if $G/\text{Z}(G)$ is cyclic.



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