Algebra I - August 2021

ALGEBRA QUALIFYING EXAM I

Problem 1.

Let $G$ be a group. In each of the following situations, either prove that $G$ is finite, or give an example where $G$ is not finite:

1. Every element of $G$ has order 2.

2. $G$ is generated by two elements $x,y\in G$, and both $x$ and $y$ have finite order.

3. $G$ is generated by two elements $x$ and $y$, and every element of $G$ has order 2.

Problem 2.

Let $p$ be a prime number. Let $G=GL_2(\mathbb Z/p^2\mathbb Z)$ denote the group of invertible $2\times2$ matrices with coefficients in $\mathbb Z/p^2\mathbb Z$. Let $V=\mathbb Z/p^2\mathbb Z\times\mathbb Z/p^2\mathbb Z$, and let

$$P(V)=\{W\subset V\mid V/W\cong\mathbb Z/p^2\mathbb Z\}$$

be the set of subgroups $W$ of $V$ such that $V/W$ is isomorphic to $\mathbb Z/p^2\mathbb Z$.

1. Compute the order of $G$. (Hint: you may use the following fact without proof: the order of $GL_2(\mathbb Z/p\mathbb Z)$ is $(p^2-1)(p^2-p)$.)

2. Show that, for any $W\in P(V)$, the group $W$ is cyclic.

3. Show that $G$ acts transitively on $P(V)$ and compute the cardinality of $P(V)$.

4. Let $G$ act on $P(V)\times P(V)$ diagonally. Show that the map

$$P(V)\times P(V)\to\{1,p,p^2\}$$

given by $(W_1,W_2)\mapsto\#(W_1\cap W_2)$ induces a bijection

$$G\backslash(P(V)\times P(V))\longleftrightarrow\{1,p,p^2\}.$$

Problem 3.

Recall that a morphism $f:A\to B$ in a category is called an epimorphism if for any morphisms $g:B\to C$ and $h:B\to C$ such that $g\circ f=h\circ f$, we have $g=h$.

1. Let $R$ be a ring. Show that the epimorphisms in the category of $R$-modules are exactly the surjective $R$-module homomorphisms.

2. Show that the group homomorphism $\mathbb Z\xrightarrow{2}\mathbb Z$ given by $x\mapsto2x$ is an epimorphism in the category of free abelian groups. (In the category of free abelian groups, objects are free abelian groups and morphisms are group homomorphisms.)

3. Show that the inclusion $\mathbb Z\to\mathbb Q$ is an epimorphism in the category of rings.

Problem 4.

This is a true/false question. For each of the following statements, either prove the statement or give an example showing that the statement is false:

1. If $R$ is a PID and $\mathfrak p\subset R$ is a prime ideal, then $R/\mathfrak p$ is a PID.

2. If $R$ is a UFD and $\mathfrak p\subset R$ is a prime ideal, then $R/\mathfrak p$ is a UFD. (Hint: you may use the following fact without proof: if $A$ is a UFD, then $A[x]$ is a UFD.)