Algebra Qual 2020-09

Let $A=\begin{bmatrix} 6 & -2 & -1 \\ 10 & -3 & -2 \\ 0 & 0 & 1\end{bmatrix}$.
\begin{enumerate}[label=\alph*)]
	\item Find bases for the eigenspaces of $A$.
	\item Determine if $A$ is diagonalizable. If so, give an invertible matrix $P$ and diagonal matrix $D$ such that $P^{-1}AP=D$. If not, explain why not.
\end{enumerate}

Let $G$ be the additive group ${\bf Z}_{2020}$ and let $H\subseteq G$ be the subset consisting of those elements with order dividing 20.
\begin{enumerate}[label=\alph*)]
	\item Prove $H$ is a subgroup of $G$.
	\item Find an explicit generator for $H$ and determine its order.
\end{enumerate}

Let $G$ be a finite group and $\operatorname{Z}(G)$ denote its center.
\begin{enumerate}[label=\alph*)]
	\item Prove that if $G/\operatorname{Z}(G)$ is cyclic, then $G$ is abelian.
	\item Prove that if $G$ is nonabelian, then $|\operatorname{Z}(G)|\leq \frac{1}{4}|G|$.
\end{enumerate}

Let $R$ be a commutative ring with 1. We say an element $n\in R$ is {\bfseries nilpotent} if there exists a number $k\in {\bf N}$ such that $n^k=0$.
\begin{enumerate}[label=\alph*)]
	\item Show that if $n$ is nilpotent, then $1-n$ is a unit.
	\item Give an example of a commutative ring with 1 that has no nonzero nilpotent elements, but is not an integral domain.
\end{enumerate}

Let $R$ be a commutative ring with 1 and suppose $e\in R$ is {\bfseries idempotent}, i.e., satisfies $e^2=e$.
\begin{enumerate}[label=\alph*)]
	\item Prove that $1-e$ is also idempotent.
	\item Suppose $e\neq 0, 1$. Show that $Re$ and $R(1-e)$ are proper ideals of $R$.
	\item Prove there is an isomorphism $R\cong Re\times R(1-e)$.
\end{enumerate}