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\begin{document}
\title{Homework $\#$5}
\author{Due 10/23/13 at start of class }
\maketitle
\subsection*{Problem 1 (D$\&$F 3.3.3)} Let $G$ be a group, $H$ a normal subgroup of $G$ with $[G:H]=p$ for some prime $p$. Prove for all subgroups $K\leq G$ that either
\begin{itemize}
\item[(i)] $K\leq H$, or
\item[(ii)] $G=HK$ and $\vert K:K\cap H\vert=p$. 
\end{itemize}  

\subsection*{Problem 2} Let $n$ and $m$ be positive integers. Let $[n,m]$ denote the least common multiple of $n$ and $m$
\begin{itemize}
\item[(a)] Use the First Isomorphism Theorem to show that there is an injective homomorphism of $\mathbb{Z}/[n, m]\mathbb{Z}$ into the direct product $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$.
\item[(b)] Show that your map in part (a) is surjective (and therefore an isomorphism) if and only if $(n, m) = 1$. Conclude that in this case $\mathbb{Z}/nm\mathbb{Z}\cong \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$.
\end{itemize}

\subsection*{Problem 3}
Let $\mathbb{Z}[x]$ denote the set of polynomials with coefficients in $\mathbb{Z}$. It is clear that $\mathbb{Z}[x]$ is an additive group. 
\begin{itemize}
\item[(a)] Define $\phi : \mathbb{Z}[x] \rightarrow \mathbb{Z}$ by $\phi(f) = f(0)$. Show that $\phi$ is a surjective homomorphism and compute its kernel. What statement does the first isomorphism theorem yield?
\item[(b)] Do the same as above with $\phi : \mathbb{Z}[x] \rightarrow \mathbb{Z}$ defined by $\phi(f) = f(3)$.
\end{itemize}

\subsection*{Problem 4 (D$\&$F 3.4.2)} Exhibit all three composition series for the quaternions $Q_8$. List the composition factors in each case.

\subsection*{Problem 5 (D$\&$F 3.4.12)} A group $G$ is \textit{solvable} if there is a chain of subgroups 
$$1=G_0\trianglelefteq G_1\trianglelefteq\cdots\trianglelefteq G_s=G$$ such that $G_{i}/G_{i-1}$ is abelian for $i=1,\dots,s$. Prove, without using the Feit-Thompson Theorem, that the following are equivalent:
\begin{itemize}
\item[(i)] every group of odd order is solvable.
\item[(ii)] the only simple groups of odd order are those of prime order.
\end{itemize}

\subsection*{Problem 6 (D$\&$F 3.5.3,4)} 
\begin{itemize}
\item[(a)] Prove that $S_n$ is generated by $\{(i\; i+1)\vert\; 1\leq i\leq n-1\}$.
\item[(b)]Prove that $S_n=\langle (12), (12\dots n)\rangle$ for $n\geq 2$.
\end{itemize}

\subsection*{Challenge Problem} Prove that the 15-puzzle arrangement given in class is unsolvable.

\subsection*{Problem 7 (D$\&$F 1.7.5)} 
For a group action of $G$ on $A$, the \textit{kernel of the action} is the set $\{g\in G\vert \; g.a=a \;\forall a\in A\}$. Prove that the kernel of the action of the group $G$ on the set $A$ is the same as the kernel of the corresponding permutation representation $\phi:G\rightarrow S_A$.

%\subsection*{Problem 8 (D$\&$F 4.1.1)} \begin{itemize}
%\item[(a)]Let $G$ act on the set $A$. Prove that if $a,b\in A$ and $b=g.a$ for some $g\in G$ then $G_b=gG_a g^{-1}$. 

%\item[(b)]An action is called \textit{transitive} if there is only one orbit, i.e., given any two elements $a,b\in A$ there is some $g\in G$ such that $a=g.b$. Deduce from part (a) that if $G$ acts transitively on $A$ then the kernel of the action is $$\bigcap_{g\in G} gG_a g^{-1}.$$
%\end{itemize}

\subsection*{Problem 8 (D$\&$F 4.2.8)}Prove that if $H$ has finite index $n$ in $G$ then there is a normal subgroup $K$ of $G$ with $K\leq H$ and $[G:K]\leq n!$.

\subsection*{Problem 9 (D\&F 4.2.10)} Prove that every non-abelian group of order $6$ has a non normal subgroup of order $2$. Use this to classify all groups of order $6$ (produce an injective homomorphism into $S_3$).

\subsection*{Problem 10 (D$\&$F 4.3.5)}If the center of $G$ is of index $n$, prove that every conjugacy class has at most $n$ elements.

\subsection*{Problem 11 (D$\&$F 4.3.6)}Assume that $G$ is a non-abelian group of order $15$. Prove that $Z(G)=1$. Use the fact that $\langle g\rangle\leq C_G(g)$ for all $g\in G$ to show there is at most one possible class equation for $G$.

\subsection*{Problem 12 (D$\&$F 4.3.7)} For $n=6$ make a list of the partitions of $n$ and give representatives for the corresponding conjugacy classes of $S_n$ (read pgs 125-128).

\end{document}