Algebra Qual 2024-01

Problem 1

Let $G$ be a group, $H\le G$ a subgroup that is not normal. Prove there exist cosets $Ha$ and $Hb$ such that $HaHb\ne Hab$.

Problem 2

Determine with proof the automorphism group $\mathrm{Aut}(V)$ of the Klein 4-group $V=\{e,a,b,ab\}$. To what familiar group is it isomorphic?

Problem 3

Suppose $R$ is a finite ring with no nontrivial zero-divisors. Prove that $R$ contains an element $1$ satisfying $1\cdot a=a\cdot 1=a$ for all $a\in R$.

Problem 4

Let $k\subset K$ be fields, and let $k[X]$ be the polynomial ring in one variable with coefficients in $k$. The evaluation at $z\in K$ is a ring homomorphism $\epsilon :k[X]\to K$ defined by $\epsilon (f(X))=f(z)$. Prove that if $\epsilon$ is not injective, then $\epsilon (k[X])$ is a field.

Problem 5

Let $V$ be a vector space with basis ${\mathbf{v}}_0,\dots ,{\mathbf{v}}_n$ and let $a_0,\dots ,a_n$ be scalars. Define a linear transformation $T:V\to V$ by the rules $T({\mathbf{v}}_i)={\mathbf{v}}_{i+1}$ if $i<n$, and $T({\mathbf{v}}_n)=a_0{\mathbf{v}}_0+a_1{\mathbf{v}}_1+\cdots +a_n{\mathbf{v}}_n$. You don't have to prove this defines a linear transformation. Determine the matrix for $T$ with respect to the basis ${\mathbf{v}}_0,\dots ,{\mathbf{v}}_n$, and determine the characteristic polynomial of $T$.