Algebra I - August 2022

ALGEBRA I QUALIFYING EXAM

AUGUST 2022

The exam will last 2 hours. You will be graded on 4 out of the 5 questions. If you submit solutions to all 5, you must state which question you choose for me to ignore on the first page of your solutions.

Problem 1.

Let $p$ be a prime. Prove that there exists a non-abelian group of order $p^3$ with normal subgroup isomorphic to $\mathbb Z/p\mathbb Z\times\mathbb Z/p\mathbb Z$.

Problem 2.

Find all isomorphism classes of abelian groups of order $10125=3^4\ast5^3$ with exactly 4 subgroups of order 15.

Problem 3.

Prove that there is no simple group of order 552.

Problem 4.

Recall that an element $r$ of a ring $R$ is nilpotent if $r^n=0$ for some $n$.

1. Show that if $R$ is commutative then the set of nilpotent elements form an ideal $I$ called the nilradical of $R$.

2. Give a counterexample if $R$ is not assumed to be commutative.

3. Give a commutative ring $R$ for which the nilradical is not prime.

Problem 5.

1. State the definition of a Noetherian module.

2. Let $M$ be a module with submodule $N$. Show that $M$ is Noetherian if and only if $N$ and $M/N$ are both Noetherian.

3. Let $F$ be the ring of real valued functions on the interval $[0,1]$. Which of the following $F$-modules are Noetherian? Justify your answers:

   1. $M_1:=\{f:[0,1]\to\mathbb R\mid f(a)=0\text{ for }a\notin\{0,1/2,1\}\}$

   2. $M_2:=\{f:[0,1]\to\mathbb R\mid f(a)=0\text{ for all but finitely many }a\in[0,1]\}$

   3. $F$ as a module over itself.