Algebra Qual 2016-09

Suppose $A$ and $B$ are subgroups of a group $G$, and suppose $B$ is of finite index in $G$.
\begin{enumerate}[topsep=0.1in]
	\item Show that the index of $A\cap B\leq A$ is finite, and in fact $|A:A\cap B|\leq |G:B|$. {\itshape Hint:} Find a set map $A/A\cap B\to G/B$.
	\item Prove that equality holds in (a) if and only if $G=AB$.
\end{enumerate}

Let $G$ be a group. Prove that $G$ is non-cyclic if and only if $G$ is the union of its proper subgroups.

\begin{enumerate}[topsep=0.1in]
	\item Write down an irreducible cubic polynomial over ${\bf F}_2$.
	\item Construct a field with exactly eight elements and write down its multiplication table.
\end{enumerate}

Consider the following matrix:
	\[
		A=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0\end{bmatrix}.
	\]
\begin{enumerate}[label=\alph*)]
	\item Determine the characteristic and minimal polynomials of $A$.
	\item Find a basis for ${\bf R}^4$ consisting of generalized eigenvectors of $A$.
	\item Find an invertible matrix $S$ such that $S^{-1}AS$ is in Jordan canonical form.
	\item Determine a Jordan canonical form of $A$.
	\end{enumerate}

Let $V$ be a vector space and $T:V\to V$ be a linear transformation.
\begin{enumerate}[label=\alph*)]
	\item Prove that if $T$ is a projection (i.e., $T^2=T$), then $V$ can be decomposed into the internal direct sum $V=\operatorname{null}(T)\oplus \operatorname{range}(T)$.
	\item Suppose $V$ is an inner product space and $T^*$ is the adjoint of $T$ with respect to the inner product. Show that $\operatorname{null}(T^*)$ is the orthogonal complement of $\operatorname{range}(T)$.
	\item Suppose $V$ is an inner product space and $T$ is an orthogonal projection, i.e., a projection for which the null space and range are orthogonal. Show that $T$ is self adjoint.
\end{enumerate}