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\title{Math 332: Homework 3}
%\author{Your name here --- delete the % at the start of the line}
\date{16.II.15}

\begin{document}
\maketitle

\begin{prob}
Prove that if $H$ and $K$ are finite subgroups of $G$ whose orders are relatively prime, then $H\cap K = 1$.
\end{prob}

\begin{prob}
Use Lagrange's Theorem in the multiplicative group $(\ZZ/p\ZZ)^\times$ to prove \emph{Fermat's little theorem}: if $p$ is  prime, then $a^p \equiv a\pmod{p}$ for all $a\in \ZZ$.
\end{prob}

\begin{prob}
Prove that if $N$ is a normal subgroup of the finite group $G$ and $(|N|,[G:N])=1$, then $N$ is the unique subgroup of $G$ of order $|N|$.
\end{prob}

\begin{prob}
Prove that if $H$ is a normal subgroup of $G$ of prime index $p$, then for all $K\le G$, either
\begin{enumerate}[(i)]
\item $K\le H$ or
\item $G = HK$ and $[K:K\cap H]=p$.
\end{enumerate}
\end{prob}

\begin{prob}
Let $p$ be a prime and let $\mu_{p^\infty}(\CC)$ be the group of $p$-power roots of unity in $\CC$.  Show that the map $z\mapsto z^p$ is a surjective homomorphism.  Deduce that $\mu_{p^\infty}(\CC)$ is isomorphic to a proper quotient of itself.  (This means that $\mu_{p^\infty}(\CC)\cong \mu_{p^\infty}(\CC)/N$ for some $1\ne N\normal \mu_{p^\infty}(\CC)$.)
\end{prob}

\begin{prob}\label{prob:univ}
Suppose that $N$ is a normal subgroup of $G$, let $i:N\to G$ denote the inclusion of $N$ into $G$, let $\pi:G\to G/N$ denote the natural projection, and $\varphi:G\to H$ be a homomorphism to a group $H$.  Consider the diagram
\[
\xymatrix{
  N\ar[r]^i\ar[dr] &G\ar[r]^\pi\ar[d]^\varphi &G/N\ar@{-->}[dl]^{\bar{\varphi}} \\
  &H
}
\]
Prove that a homomorphism $\bar{\varphi}:G/N\to H$ making the diagram commute exists if and only if $\varphi\circ i$ is the trivial homomorphism.  (Note that the condition $\varphi\circ i=1$ is equivalent to $\varphi(N)=1$, which is in turn equivalent to $\ker\varphi\le N$.)  Show additionally that when such a $\bar{\varphi}$ exists, it is unique.  \emph{Bonus}: Show that this property of $\pi:G\to G/N$ uniquely characterizes it.  In other words, show that if $p:G\to K$ is any other group homomorphism satisfying the same ``unique extension" property, then there is a unique isomorphism $\Phi:G/N\to K$ such that $\Phi\circ \pi = p$.  (This justifies the moniker under which the above property goes:  the \emph{universal property} of the quotient map $G\to G/N$.)
\end{prob}

\begin{prob}
Let $M$ and $N$ be normal subgroups of $G$ such that $G = MN$.  Prove that $G/(M\cap N)\cong (G/M)\times (G/N)$.  (You may want to use your result from Problem \ref{prob:univ}.)
\end{prob}

\begin{prob}
Prove that subgroups and quotient groups of a solvable group are solvable.
\end{prob}

\begin{prob}
Prove that $\sigma^2$ is an even permutation for any permutation $\sigma$.
\end{prob}

\begin{prob}
Show that $S_n = \langle (1\enspace 2), (1\enspace 2\enspace 3\enspace \cdots\enspace n)\rangle$ for all $n\ge 2$.
\end{prob}

\begin{prob}
Prove that the group of rigid motions of the tetrahedron is isomorphic to $A_4$.
\end{prob}

\end{document}