Algebra Qual 2024-09

Problem 1

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.

Problem 2

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

Problem 3

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

Problem 4

Let $T:V\to V$ be a linear transformation on a finite-dimensional vector space. Prove that if $T^2=T$, then

Problem 5

Let ${\mathbf{R}}^3$ denote the $3$-dimensional vector space, and let $\mathbf{v}=(a,b,c)$ be a fixed nonzero vector. The maps $C:{\mathbf{R}}^3\to {\mathbf{R}}^3$ and $D:{\mathbf{R}}^3\to \mathbf{R}$ defined by $C(\mathbf{w})=\mathbf{v}\times \mathbf{w}$ and $D(\mathbf{w})=(\mathbf{v}\cdot \mathbf{w})\mathbf{v}$ are linear transformations.
(a) Determine the eigenvalues of $C$ and $D$.
(b) Determine the eigenspaces of $C$ and $D$ as subspaces of ${\mathbf{R}}^3$, in terms of $a,b,c$.
(c) Find a matrix for $C$ with respect to the standard basis.

Show all work and explain reasoning.