Algebra Qual 2015-09

Determine the number of group homomorphisms $\phi$ between the given groups. Here $K_4$ denotes the Klein four-group (also known as ${\bf Z}/2{\bf Z}\times {\bf Z}/2{\bf Z}$) and $S_3$ denotes the symmetric group on three elements.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item $\phi:K_4\to {\bf Z}/2{\bf Z}$
	\item $\phi:{\bf Z}/2{\bf Z}\to K_4$
	\item $\phi:S_3\to K_4$
	\item $\phi:K_4\to S_3$
\end{enumerate}

\begin{enumerate}[label=(\alph*)]
	\item Show that if $G$ is any group (not necessarily finite) and $H$ is a subgroup, then $G$ is a disjoint union of left cosets of $H$.
	\item State and prove Lagrange's Theorem for finite groups.
\end{enumerate}

Let $R$ be an integral domain. Suppose that $a$ and $b$ are non-associate irreducible elements in $R$, and the ideal $(a,b)$ generated by $a$ and $b$ is a proper ideal. Show that $R$ is not a principal ideal domain (PID).

Let $F$ be a field and let $\alpha$ be an element that generates a field extension of $F$ of degree five. Prove that $\alpha^2$ generates the same extension.

Let $A=\begin{bmatrix} -2 & 1 & -1 \\ 5 & -2 & 2 \\ 7 & -3 & 3\end{bmatrix}$.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Find the characteristic polynomial and the minimal polynomial of $A$.
	\item Find the Jordan canonical form of the matrix $A$.
\end{enumerate}