Algebra Qual 2024-09

Let $G$ be a group, and $G\times G$ the direct product. The set $D=\{(g,g,)\mid g\in G\}$ is a subgroup of $G\times G$. Prove that if $D$ is normal in $G\times G$ then $G$ is abelian.

 Let $G$ be a group with exactly two conjugacy classes. Prove that $G$ is abelian, and describe all such groups (with proof).

Suppose $\phi:R\to S$ is a ring homomorphism, and $S$ has no (nonzero) zero-divisors. Prove from the definitions that $\ker(\phi)$ is a prime ideal.

Let $T:V\to V$ be a linear transformation on a finite-dimensional vector space. Prove that if $T^2=T$, then
\[
	V=\ker(T)\oplus \operatorname{im}(T).
\]

Let ${\bf R}^3$ denote the $3$-dimensional vector space, and let ${\bf v}=(a,b,c)$ be a fixed nonzero vector. The maps $C:{\bf R}^3\to {\bf R}^3$ and $D:{\bf R}^3\to {\bf R}$ defined by $C({\bf w})={\bf v}\times {\bf w}$ and $D({\bf w})=({\bf v}\cdot {\bf w}){\bf v}$ are linear transformations.
	\begin{enumerate}[label=\alph*)]
		\item Determine the eigenvalues of $C$ and $D$.
		
		\item Determine the eigenspaces of $C$ and $D$ as subspaces of ${\bf R}^3$, in terms of $a, b, c$.
		
		\item Find a matrix for $C$ with respect to the standard basis. 
	\end{enumerate}
Show all work and explain reasoning.