Algebra Qual 2025-01

Let $G$ be a group, and $H, K$ be subgroups of $G$. Let $HK=\{hk\,\mid \, h\in H, k\in K\}$ denote the set product. Prove that $HK$ is a group if and only if $HK=KH$.

Suppose $\phi:G\to G'$ is a surjective homomorphism, $H\leq G$ is a subgroup containing $\ker(\phi)$, and $H'=\phi(H)$. Prove $\phi^{-1}(H')=H$, where $\phi^{-1}(H')=\{g\in G\,\mid\, \phi(g)\in H'\}$. Make sure to state explicitly where each hypothesis is used.

A {\bfseries Boolean algebra} is a ring $A$ with $1$ satisfying $x^2=x$ for all $x\in A$. Prove that in a Boolean algebra $A$:
	\begin{enumerate}[label=\alph*)]
		\item $2x=0$ for all $x\in A$.
		
		\item Every prime ideal $\mathfrak{p}$ is maximal, and $A/\mathfrak{p}$ is a field with two elements.
	\end{enumerate}

Let ${\bf Z}_n$ be the ring of integers $\pmod{n}$. There is a ring homomorphism
	\begin{align*}
		{\bf Z}_{28}&\to {\bf Z}_4\times {\bf Z}_7\\
		[m]_{28}&\mapsto ([m]_4,[m]_7)
	\end{align*}
This is an isomorphism by the Chinese Remainder Theorem. Let ${\bf Z}_n^{\times}$ be the group of units of ${\bf Z}_n$. Prove that ${\bf Z}_{28}^{\times}$ is isomorphic to ${\bf Z}_4^{\times}\times {\bf Z}_7^{\times}$.

Let $T:{\bf R}^3\to {\bf R}^3$ be the orthogonal projection onto the plane $z=x+y$, with respect to the standard Euclidean inner product.
	\begin{enumerate}[label=\alph*)]
		\item Write the matrix representation of $T$ with respect to the standard basis.
		
		\item Is $T$ diagonalizable? Justify your answer.
	\end{enumerate}