Algebra Qual 2025-01

Problem 1

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 $HK=KH$.

Problem 2

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

Problem 3

A Boolean algebra is a ring $A$ with $1$ satisfying $x^2=x$ for all $x\in A$. Prove that in a Boolean algebra $A$:
(a) $2x=0$ for all $x\in A$.
(b) Every nonzero prime ideal $\mathfrak{p}$ is maximal, and $A/\mathfrak{p}$ is a field with two elements.

Problem 4

Let ${\mathbf{Z}}_n$ be the ring of integers $(\mathrm{mod}n)$. There is a ring homomorphism

This is an isomorphism by the Chinese Remainder Theorem. Let ${\mathbf{Z}}_n^{\times }$ be the group of units of ${\mathbf{Z}}_n$. Prove that ${\mathbf{Z}}_{28}^{\times }$ is isomorphic to ${\mathbf{Z}}_4^{\times }\times {\mathbf{Z}}_7^{\times }$.

Problem 5

Let $T:{\mathbf{R}}^3\to {\mathbf{R}}^3$ be the orthogonal projection onto the plane $z=x+y$, with respect to the standard Euclidean inner product.
(a) Write the matrix representation of $T$ with respect to the standard basis.
(b) Is $T$ diagonalizable? Justify your answer.