Algebra Qual 2016-03

Problem 1

Without using Cauchy's Theorem or the Sylow theorems, prove that every group of order 21 contains an element of order three.

Problem 2

Suppose $G$ is a group that contains normal subgroups $H,K⊴G$ with $H\cap K=\{e\}$ and $HK=G$. Prove that $G\cong H\times K$.

Problem 3

Let $R$ be a commutative ring.
(a) Prove that the set $N$ of all nilpotent elements of $R$ is an ideal.
(b) Prove that $R/N$ is a ring with no nonzero nilpotent elements.
(c) Show that $N$ is contained in every prime ideal of $R$.

Problem 4

Let $z\in \mathbf{C}$ be a complex number and let ${ϵ}_z:\mathbf{R}[x]\to \mathbf{C}$ be the evaluation homomorphism given by ${ϵ}_z(p)=p(z)$ for each $p\in \mathbf{R}[x]$.
(a) Show that $\ker ({ϵ}_z)$ is a prime ideal.
(b) Compute $\ker ({ϵ}_{1+i}),\mathrm{im}({ϵ}_{1+i})$ and then state the conclusion of the First Isomorphism Theorem applied to the homomorphism ${ϵ}_{1+i}$.

Problem 5

Let $T:{\mathbf{R}}^3\to {\mathbf{R}}^3$ be the linear transformation that expands radially by a factor of three around the line parameterized by $L(t)=\left[\begin{array}{c}2\\ 2\\ -1\end{array}\right]t$, leaving the line itself fixed (viewed as a subspace).
(a) Find an eigenbasis for $T$ and provide the matrix representation of $T$ with respect to that basis.
(b) Provide the matrix representation of $T$ with respect to the standard basis.