Algebra I - August 2024

ALGEBRA I QUALIFYING EXAM AUGUST 2024

Problem 1.

Let $H$ be a proper subgroup of a finite group $G$.

1. Prove that for any $g\in G$, the subgroup $gHg^{-1}$ of $G$ is isomorphic to $H$.

2. Prove that as $g$ ranges over the elements of $G$, the number of distinct sets $gHg^{-1}$ is at most the index of $H$ in $G$.

3. Conclude that $G\ne\bigcup_{g\in G}gHg^{-1}$.

Problem 2.

Prove that there is no simple group of order 280.

Problem 3.

Let $R$ be a commutative ring and $P,Q$ prime ideals in $R$. Prove that $\operatorname{Hom}_R(R/P,R/Q)$ is nonzero if and only if $P\subseteq Q$.

Problem 4.

1. Let $S$ be a commutative ring with identity and $\tau$ a unital ring automorphism of $S$. Show that the image of a principal ideal of $S$ under $\tau$ is principal and that the image of a prime ideal is prime.

2. Consider the principal ideal $I=(y^2-x^3+x)$ in $\mathbb Q[x,y]$ and let $R=\mathbb Q[x,y]/I$.

   1. Show that the automorphism of $\mathbb Q[x,y]$ given by $x\mapsto x$ and $y\mapsto-y$ descends to an automorphism $\sigma$ of $R$.

   2. Show that the ideal $P=(x,y)R$ in $R$ is a prime ideal.

   3. Show that $P^2$ is a principal ideal.

Problem 5.

Let $R$ be a commutative ring with identity.

1. Prove that a module $M$ over $R$ is projective if and only if $M$ is a direct summand of a free module.

2. Prove that a projective module over an integral domain is torsion free.

3. Let $0\to M_1\to M_2\to M_3\to0$ be a short exact sequence of finitely-generated $R$-modules. Either prove or give a counter-example to the following.

   1. If $M_1$ and $M_2$ are projective, then so is $M_3$.

   2. If $M_2$ and $M_3$ are projective, then so is $M_1$.

Problem 6.

Let $\mathbb F_2$ be the field with 2 elements and consider the ring $R=\mathbb F_2[x]$. List, up to isomorphism, all $R$-modules with 8 elements.