# Groups, Rings And Modules

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Paper 1, Section II, G

commentShow that a ring $R$ is Noetherian if and only if every ideal of $R$ is finitely generated. Show that if $\phi: R \rightarrow S$ is a surjective ring homomorphism and $R$ is Noetherian, then $S$ is Noetherian.

State and prove Hilbert's Basis Theorem.

Let $\alpha \in \mathbb{C}$. Is $\mathbb{Z}[\alpha]$ Noetherian? Justify your answer.

Give, with proof, an example of a Unique Factorization Domain that is not Noetherian.

Let $R$ be the ring of continuous functions $\mathbb{R} \rightarrow \mathbb{R}$. Is $R$ Noetherian? Justify your answer.

Paper 2, Section I, $1 G$

commentLet $M$ be a module over a Principal Ideal Domain $R$ and let $N$ be a submodule of $M$. Show that $M$ is finitely generated if and only if $N$ and $M / N$ are finitely generated.

Paper 2, Section II, G

commentLet $M$ be a module over a ring $R$ and let $S \subset M$. Define what it means that $S$ freely generates $M$. Show that this happens if and only if for every $R$-module $N$, every function $f: S \rightarrow N$ extends uniquely to a homomorphism $\phi: M \rightarrow N$.

Let $M$ be a free module over a (non-trivial) ring $R$ that is generated (not necessarily freely) by a subset $T \subset M$ of size $m$. Show that if $S$ is a basis of $M$, then $S$ is finite with $|S| \leqslant m$. Hence, or otherwise, deduce that any two bases of $M$ have the same number of elements. Denoting this number $\operatorname{rk} M$ and by quoting any result you need, show that if $R$ is a Euclidean Domain and $N$ is a submodule of $M$, then $N$ is free with $\operatorname{rk} N \leqslant \operatorname{rk} M$.

State the Primary Decomposition Theorem for a finitely generated module $M$ over a Euclidean Domain $R$. Deduce that any finite subgroup of the multiplicative group of a field is cyclic.

Paper 3, Section I, G

commentLet $G$ be a finite group, and let $H$ be a proper subgroup of $G$ of index $n$.

Show that there is a normal subgroup $K$ of $G$ such that $|G / K|$ divides $n$ ! and $|G / K| \geqslant n$.

Show that if $G$ is non-abelian and simple, then $G$ is isomorphic to a subgroup of $A_{n}$.

Paper 3, Section II, 10G

commentLet $p$ be a non-zero element of a Principal Ideal Domain $R$. Show that the following are equivalent:

(i) $p$ is prime;

(ii) $p$ is irreducible;

(iii) $(p)$ is a maximal ideal of $R$;

(iv) $R /(p)$ is a field;

(v) $R /(p)$ is an Integral Domain.

Let $R$ be a Principal Ideal Domain, $S$ an Integral Domain and $\phi: R \rightarrow S$ a surjective ring homomorphism. Show that either $\phi$ is an isomorphism or $S$ is a field.

Show that if $R$ is a commutative ring and $R[X]$ is a Principal Ideal Domain, then $R$ is a field.

Let $R$ be an Integral Domain in which every two non-zero elements have a highest common factor. Show that in $R$ every irreducible element is prime.

Paper 4, Section II, G

commentLet $H$ and $P$ be subgroups of a finite group $G$. Show that the sets $H x P, x \in G$, partition $G$. By considering the action of $H$ on the set of left cosets of $P$ in $G$ by left multiplication, or otherwise, show that

$\frac{|H x P|}{|P|}=\frac{|H|}{\left|H \cap x P x^{-1}\right|}$

for any $x \in G$. Deduce that if $G$ has a Sylow $p$-subgroup, then so does $H$.

Let $p, n \in \mathbb{N}$ with $p$ a prime. Write down the order of the group $G L_{n}(\mathbb{Z} / p \mathbb{Z})$. Identify in $G L_{n}(\mathbb{Z} / p \mathbb{Z})$ a Sylow $p$-subgroup and a subgroup isomorphic to the symmetric group $S_{n}$. Deduce that every finite group has a Sylow $p$-subgroup.

State Sylow's theorem on the number of Sylow $p$-subgroups of a finite group.

Let $G$ be a group of order $p q$, where $p>q$ are prime numbers. Show that if $G$ is non-abelian, then $q \mid p-1$.

Paper 1, Section II, G

commentState the structure theorem for a finitely generated module $M$ over a Euclidean domain $R$ in terms of invariant factors.

Let $V$ be a finite-dimensional vector space over a field $F$ and let $\alpha: V \rightarrow V$ be a linear map. Let $V_{\alpha}$ denote the $F[X]$-module $V$ with $X$ acting as $\alpha$. Apply the structure theorem to $V_{\alpha}$ to show the existence of a basis of $V$ with respect to which $\alpha$ has the rational canonical form. Prove that the minimal polynomial and the characteristic polynomial of $\alpha$ can be expressed in terms of the invariant factors. [Hint: For the characteristic polynomial apply suitable row operations.] Deduce the Cayley-Hamilton theorem for $\alpha$.

Now assume that $\alpha$ has matrix $\left(a_{i j}\right)$ with respect to the basis $v_{1}, \ldots, v_{n}$ of $V$. Let $M$ be the free $F[X]$-module of rank $n$ with free basis $m_{1}, \ldots, m_{n}$ and let $\theta: M \rightarrow V_{\alpha}$ be the unique homomorphism with $\theta\left(m_{i}\right)=v_{i}$ for $1 \leqslant i \leqslant n$. Using the fact, which you need not prove, that ker $\theta$ is generated by the elements $X m_{i}-\sum_{j=1}^{n} a_{j i} m_{j}, 1 \leqslant i \leqslant n$, find the invariant factors of $V_{\alpha}$ in the case that $V=\mathbb{R}^{3}$ and $\alpha$ is represented by the real matrix

$\left(\begin{array}{ccc} 0 & 1 & 0 \\ -4 & 4 & 0 \\ -2 & 1 & 2 \end{array}\right)$

with respect to the standard basis.

Paper 2, Section I, G

commentAssume a group $G$ acts transitively on a set $\Omega$ and that the size of $\Omega$ is a prime number. Let $H$ be a normal subgroup of $G$ that acts non-trivially on $\Omega$.

Show that any two $H$-orbits of $\Omega$ have the same size. Deduce that the action of $H$ on $\Omega$ is transitive.

Let $\alpha \in \Omega$ and let $G_{\alpha}$ denote the stabiliser of $\alpha$ in $G$. Show that if $H \cap G_{\alpha}$ is trivial, then there is a bijection $\theta: H \rightarrow \Omega$ under which the action of $G_{\alpha}$ on $H$ by conjugation corresponds to the action of $G_{\alpha}$ on $\Omega$.

Paper 2, Section II, G

commentState Gauss' lemma. State and prove Eisenstein's criterion.

Define the notion of an algebraic integer. Show that if $\alpha$ is an algebraic integer, then $\{f \in \mathbb{Z}[X]: f(\alpha)=0\}$ is a principal ideal generated by a monic, irreducible polynomial.

Let $f=X^{4}+2 X^{3}-3 X^{2}-4 X-11$. Show that $\mathbb{Q}[X] /(f)$ is a field. Show that $\mathbb{Z}[X] /(f)$ is an integral domain, but not a field. Justify your answers.

Paper 1, Section II, G

comment(a) Let $G$ be a group of order $p^{4}$, for $p$ a prime. Prove that $G$ is not simple.

(b) State Sylow's theorems.

(c) Let $G$ be a group of order $p^{2} q^{2}$, where $p, q$ are distinct odd primes. Prove that $G$ is not simple.

Paper 2, Section I, G

commentLet $R$ be an integral domain. A module $M$ over $R$ is torsion-free if, for any $r \in R$ and $m \in M, r m=0$ only if $r=0$ or $m=0$.

Let $M$ be a module over $R$. Prove that there is a quotient

$q: M \rightarrow M_{0}$

with $M_{0}$ torsion-free and with the following property: whenever $N$ is a torsion-free module and $f: M \rightarrow N$ is a homomorphism of modules, there is a homomorphism $f_{0}: M_{0} \rightarrow N$ such that $f=f_{0} \circ q$.

Paper 2, Section II, G

comment(a) Let $k$ be a field and let $f(X)$ be an irreducible polynomial of degree $d>0$ over $k$. Prove that there exists a field $F$ containing $k$ as a subfield such that

$f(X)=(X-\alpha) g(X)$

where $\alpha \in F$ and $g(X) \in F[X]$. State carefully any results that you use.

(b) Let $k$ be a field and let $f(X)$ be a monic polynomial of degree $d>0$ over $k$, which is not necessarily irreducible. Prove that there exists a field $F$ containing $k$ as a subfield such that

$f(X)=\prod_{i=1}^{d}\left(X-\alpha_{i}\right)$

where $\alpha_{i} \in F$.

(c) Let $k=\mathbb{Z} /(p)$ for $p$ a prime, and let $f(X)=X^{p^{n}}-X$ for $n \geqslant 1$ an integer. For $F$ as in part (b), let $K$ be the set of roots of $f(X)$ in $F$. Prove that $K$ is a field.

Paper 3, Section I, $1 G$

commentProve that the ideal $(2,1+\sqrt{-13})$ in $\mathbb{Z}[\sqrt{-13}]$ is not principal.

Paper 3, Section II, G

commentLet $\omega=\frac{1}{2}(-1+\sqrt{-3})$.

(a) Prove that $\mathbb{Z}[\omega]$ is a Euclidean domain.

(b) Deduce that $\mathbb{Z}[\omega]$ is a unique factorisation domain, stating carefully any results from the course that you use.

(c) By working in $\mathbb{Z}[\omega]$, show that whenever $x, y \in \mathbb{Z}$ satisfy

$x^{2}-x+1=y^{3}$

then $x$ is not congruent to 2 modulo 3 .

Paper 4, Section I, G

commentLet $G$ be a group and $P$ a subgroup.

(a) Define the normaliser $N_{G}(P)$.

(b) Suppose that $K \triangleleft G$ and $P$ is a Sylow $p$-subgroup of $K$. Using Sylow's second theorem, prove that $G=N_{G}(P) K$.

Paper 4, Section II, G

comment(a) Define the Smith Normal Form of a matrix. When is it guaranteed to exist?

(b) Deduce the classification of finitely generated abelian groups.

(c) How many conjugacy classes of matrices are there in $G L_{10}(\mathbb{Q})$ with minimal polynomial $X^{7}-4 X^{3} ?$

Paper 1, Section II, G

comment(a) State Sylow's theorems.

(b) Prove Sylow's first theorem.

(c) Let $G$ be a group of order 12. Prove that either $G$ has a unique Sylow 3-subgroup or $G \cong A_{4}$.

Paper 2, Section $I$, $2 G$

commentLet $R$ be a principal ideal domain and $x$ a non-zero element of $R$. We define a new $\operatorname{ring} R^{\prime}$ as follows. We define an equivalence relation $\sim$ on $R \times\left\{x^{n} \mid n \in \mathbb{Z}_{\geqslant 0}\right\}$ by

$\left(r, x^{n}\right) \sim\left(r^{\prime}, x^{n^{\prime}}\right)$

if and only if $x^{n^{\prime}} r=x^{n} r^{\prime}$. The underlying set of $R^{\prime}$ is the set of $\sim$-equivalence classes. We define addition on $R^{\prime}$ by

$\left[\left(r, x^{n}\right)\right]+\left[\left(r^{\prime}, x^{n^{\prime}}\right)\right]=\left[\left(x^{n^{\prime}} r+x^{n} r^{\prime}, x^{n+n^{\prime}}\right)\right]$

and multiplication by $\left[\left(r, x^{n}\right)\right]\left[\left(r^{\prime}, x^{n^{\prime}}\right)\right]=\left[\left(r r^{\prime}, x^{n+n^{\prime}}\right)\right]$.

(a) Show that $R^{\prime}$ is a well defined ring.

(b) Prove that $R^{\prime}$ is a principal ideal domain.

Paper 2, Section II, G

comment(a) Prove that every principal ideal domain is a unique factorization domain.

(b) Consider the ring $R=\{f(X) \in \mathbb{Q}[X] \mid f(0) \in \mathbb{Z}\}$.

(i) What are the units in $R$ ?

(ii) Let $f(X) \in R$ be irreducible. Prove that either $f(X)=\pm p$, for $p \in \mathbb{Z}$ a prime, or $\operatorname{deg}(f) \geqslant 1$ and $f(0)=\pm 1$.

(iii) Prove that $f(X)=X$ is not expressible as a product of irreducibles.

Paper 3, Section I, $1 G$

comment(a) Find all integer solutions to $x^{2}+5 y^{2}=9$.

(b) Find all the irreducibles in $\mathbb{Z}[\sqrt{-5}]$ of norm 9 .

Paper 3, Section II, G

comment(a) State Gauss's Lemma.

(b) State and prove Eisenstein's criterion for the irreducibility of a polynomial.

(c) Determine whether or not the polynomial

$f(X)=2 X^{3}+19 X^{2}-54 X+3$

is irreducible over $\mathbb{Q}$.

Paper 4, Section I, G

comment(a) Show that every automorphism $\alpha$ of the dihedral group $D_{6}$ is equal to conjugation by an element of $D_{6}$; that is, there is an $h \in D_{6}$ such that

$\alpha(g)=h g h^{-1}$

for all $g \in D_{6}$.

(b) Give an example of a non-abelian group $G$ with an automorphism which is not equal to conjugation by an element of $G$.

Paper 4, Section II, G

comment(a) State the classification theorem for finitely generated modules over a Euclidean domain.

(b) Deduce the existence of the rational canonical form for an $n \times n$ matrix $A$ over a field $F$.

(c) Compute the rational canonical form of the matrix

$A=\left(\begin{array}{ccc} 3 / 2 & 1 & 0 \\ -1 & -1 / 2 & 0 \\ 2 & 2 & 1 / 2 \end{array}\right)$

Paper 1, Section II, 10E

comment(a) State Sylow's theorem.

(b) Let $G$ be a finite simple non-abelian group. Let $p$ be a prime number. Show that if $p$ divides $|G|$, then $|G|$ divides $n_{p} ! / 2$ where $n_{p}$ is the number of Sylow $p$-subgroups of $G$.

(c) Let $G$ be a group of order 48 . Show that $G$ is not simple. Find an example of $G$ which has no normal Sylow 2-subgroup.

Paper 2, Section I, E

comment(a) Define what is meant by a unique factorisation domain and by a principal ideal domain. State Gauss's lemma and Eisenstein's criterion, without proof.

(b) Find an example, with justification, of a ring $R$ and a subring $S$ such that

(i) $R$ is a principal ideal domain, and

(ii) $S$ is a unique factorisation domain but not a principal ideal domain.

Paper 2, Section II, E

commentLet $R$ be a commutative ring.

(a) Let $N$ be the set of nilpotent elements of $R$, that is,

$N=\left\{r \in R \mid r^{n}=0 \text { for some } n \in \mathbb{N}\right\}$

Show that $N$ is an ideal of $R$.

(b) Assume $R$ is Noetherian and assume $S \subset R$ is a non-empty subset such that if $s, t \in S$, then $s t \in S$. Let $I$ be an ideal of $R$ disjoint from $S$. Show that there is a prime ideal $P$ of $R$ containing $I$ and disjoint from $S$.

(c) Again assume $R$ is Noetherian and let $N$ be as in part (a). Let $\mathcal{P}$ be the set of all prime ideals of $R$. Show that

$N=\bigcap_{P \in \mathcal{P}} P$

Paper 3, Section I, E

commentLet $R$ be a commutative ring and let $M$ be an $R$-module. Show that $M$ is a finitely generated $R$-module if and only if there exists a surjective $R$-module homomorphism $R^{n} \rightarrow M$ for some $n$.

Find an example of a $\mathbb{Z}$-module $M$ such that there is no surjective $\mathbb{Z}$-module homomorphism $\mathbb{Z} \rightarrow M$ but there is a surjective $\mathbb{Z}$-module homomorphism $\mathbb{Z}^{2} \rightarrow M$ which is not an isomorphism. Justify your answer.

Paper 3, Section II, E

comment(a) Define what is meant by a Euclidean domain. Show that every Euclidean domain is a principal ideal domain.

(b) Let $p \in \mathbb{Z}$ be a prime number and let $f \in \mathbb{Z}[x]$ be a monic polynomial of positive degree. Show that the quotient ring $\mathbb{Z}[x] /(p, f)$ is finite.

(c) Let $\alpha \in \mathbb{Z}[\sqrt{-1}]$ and let $P$ be a non-zero prime ideal of $\mathbb{Z}[\alpha]$. Show that the quotient $\mathbb{Z}[\alpha] / P$ is a finite ring.

Paper 4, Section I, $2 E$

commentLet $G$ be a non-trivial finite $p$-group and let $Z(G)$ be its centre. Show that $|Z(G)|>1$. Show that if $|G|=p^{3}$ and if $G$ is not abelian, then $|Z(G)|=p$.

Paper 4, Section II, E

comment(a) State (without proof) the classification theorem for finitely generated modules over a Euclidean domain. Give the statement and the proof of the rational canonical form theorem.

(b) Let $R$ be a principal ideal domain and let $M$ be an $R$-submodule of $R^{n}$. Show that $M$ is a free $R$-module.

Paper 1, Section II, E

comment(a) Let $I$ be an ideal of a commutative ring $R$ and assume $I \subseteq \bigcup_{i=1}^{n} P_{i}$ where the $P_{i}$ are prime ideals. Show that $I \subseteq P_{i}$ for some $i$.

(b) Show that $\left(x^{2}+1\right)$ is a maximal ideal of $\mathbb{R}[x]$. Show that the quotient ring $\mathbb{R}[x] /\left(x^{2}+1\right)$ is isomorphic to $\mathbb{C} .$

(c) For $a, b \in \mathbb{R}$, let $I_{a, b}$ be the ideal $(x-a, y-b)$ in $\mathbb{R}[x, y]$. Show that $I_{a, b}$ is a maximal ideal. Find a maximal ideal $J$ of $\mathbb{R}[x, y]$ such that $J \neq I_{a, b}$ for any $a, b \in \mathbb{R}$. Justify your answers.

Paper 2, Section I, E

commentLet $R$ be an integral domain.

Define what is meant by the field of fractions $F$ of $R$. [You do not need to prove the existence of $F$.]

Suppose that $\phi: R \rightarrow K$ is an injective ring homomorphism from $R$ to a field $K$. Show that $\phi$ extends to an injective ring homomorphism $\Phi: F \rightarrow K$.

Give an example of $R$ and a ring homomorphism $\psi: R \rightarrow S$ from $R$ to a ring $S$ such that $\psi$ does not extend to a ring homomorphism $F \rightarrow S$.

Paper 2, Section II, E

comment(a) State Sylow's theorems and give the proof of the second theorem which concerns conjugate subgroups.

(b) Show that there is no simple group of order 351 .

(c) Let $k$ be the finite field $\mathbb{Z} /(31)$ and let $G L_{2}(k)$ be the multiplicative group of invertible $2 \times 2$ matrices over $k$. Show that every Sylow 3-subgroup of $G L_{2}(k)$ is abelian.

Paper 3, Section I, E

commentLet $G$ be a group of order $n$. Define what is meant by a permutation representation of $G$. Using such representations, show $G$ is isomorphic to a subgroup of the symmetric group $S_{n}$. Assuming $G$ is non-abelian simple, show $G$ is isomorphic to a subgroup of $A_{n}$. Give an example of a permutation representation of $S_{3}$ whose kernel is $A_{3}$.

Paper 3, Section II, E

comment(a) Define what is meant by an algebraic integer $\alpha$. Show that the ideal

$I=\{h \in \mathbb{Z}[x] \mid h(\alpha)=0\}$

in $\mathbb{Z}[x]$ is generated by a monic irreducible polynomial $f$. Show that $\mathbb{Z}[\alpha]$, considered as a $\mathbb{Z}$-module, is freely generated by $n$ elements where $n=\operatorname{deg} f$.

(b) Assume $\alpha \in \mathbb{C}$ satisfies $\alpha^{5}+2 \alpha+2=0$. Is it true that the ideal (5) in $\mathbb{Z}[\alpha]$ is a prime ideal? Is there a ring homomorphism $\mathbb{Z}[\alpha] \rightarrow \mathbb{Z}[\sqrt{-1}]$ ? Justify your answers.

(c) Show that the only unit elements of $\mathbb{Z}[\sqrt{-5}]$ are 1 and $-1$. Show that $\mathbb{Z}[\sqrt{-5}]$ is not a UFD.

Paper 4, Section I, $2 \mathrm{E}$

commentGive the statement and the proof of Eisenstein's criterion. Use this criterion to show $x^{p-1}+x^{p-2}+\cdots+1$ is irreducible in $\mathbb{Q}[x]$ where $p$ is a prime.

Paper 4, Section II, E

commentLet $R$ be a Noetherian ring and let $M$ be a finitely generated $R$-module.

(a) Show that every submodule of $M$ is finitely generated.

(b) Show that each maximal element of the set

$\mathcal{A}=\{\operatorname{Ann}(m) \mid 0 \neq m \in M\}$

is a prime ideal. [Here, maximal means maximal with respect to inclusion, and $\operatorname{Ann}(m)=\{r \in R \mid r m=0\} .]$

(c) Show that there is a chain of submodules

$0=M_{0} \subseteq M_{1} \subseteq \cdots \subseteq M_{l}=M$

such that for each $0<i \leqslant l$ the quotient $M_{i} / M_{i-1}$ is isomorphic to $R / P_{i}$ for some prime ideal $P_{i}$.

Paper 1, Section II, F

comment(i) Give the definition of a $p$-Sylow subgroup of a group.

(ii) Let $G$ be a group of order $2835=3^{4} \cdot 5 \cdot 7$. Show that there are at most two possibilities for the number of 3-Sylow subgroups, and give the possible numbers of 3-Sylow subgroups.

(iii) Continuing with a group $G$ of order 2835 , show that $G$ is not simple.

Paper 2, Section $I$, $2 F$

commentGive four non-isomorphic groups of order 12 , and explain why they are not isomorphic.

Paper 2, Section II, F

comment(a) Consider the homomorphism $f: \mathbb{Z}^{3} \rightarrow \mathbb{Z}^{4}$ given by

$f(a, b, c)=(a+2 b+8 c, 2 a-2 b+4 c,-2 b+12 c, 2 a-4 b+4 c)$

Describe the image of this homomorphism as an abstract abelian group. Describe the quotient of $\mathbb{Z}^{4}$ by the image of this homomorphism as an abstract abelian group.

(b) Give the definition of a Euclidean domain.

Fix a prime $p$ and consider the subring $R$ of the rational numbers $\mathbb{Q}$ defined by

$R=\{q / r \mid \operatorname{gcd}(p, r)=1\}$

where 'gcd' stands for the greatest common divisor. Show that $R$ is a Euclidean domain.

Paper 3, Section I, F

commentState two equivalent conditions for a commutative ring to be Noetherian, and prove they are equivalent. Give an example of a ring which is not Noetherian, and explain why it is not Noetherian.

Paper 3, Section II, F

commentCan a group of order 55 have 20 elements of order 11? If so, give an example. If not, give a proof, including the proof of any statements you need.

Let $G$ be a group of order $p q$, with $p$ and $q$ primes, $p>q$. Suppose furthermore that $q$ does not divide $p-1$. Show that $G$ is cyclic.

Paper 4, Section I, $2 F$

commentLet $R$ be a commutative ring. Define what it means for an ideal $I \subseteq R$ to be prime. Show that $I \subseteq R$ is prime if and only if $R / I$ is an integral domain.

Give an example of an integral domain $R$ and an ideal $I \subset R, I \neq R$, such that $R / I$ is not an integral domain.

Paper 4, Section II, F

commentFind $a \in \mathbb{Z}_{7}$ such that $\mathbb{Z}_{7}[x] /\left(x^{3}+a\right)$ is a field $F$. Show that for your choice of $a$, every element of $\mathbb{Z}_{7}$ has a cube root in the field $F$.

Show that if $F$ is a finite field, then the multiplicative group $F^{\times}=F \backslash\{0\}$ is cyclic.

Show that $F=\mathbb{Z}_{2}[x] /\left(x^{3}+x+1\right)$ is a field. How many elements does $F$ have? Find a generator for $F^{\times}$.

Paper 1, Section II, E

commentLet $G$ be a finite group and $p$ a prime divisor of the order of $G$. Give the definition of a Sylow $p$-subgroup of $G$, and state Sylow's theorems.

Let $p$ and $q$ be distinct primes. Prove that a group of order $p^{2} q$ is not simple.

Let $G$ be a finite group, $H$ a normal subgroup of $G$ and $P$ a Sylow $p$-subgroup of H. Let $N_{G}(P)$ denote the normaliser of $P$ in $G$. Prove that if $g \in G$ then there exist $k \in N_{G}(P)$ and $h \in H$ such that $g=k h$.

Paper 2, Section I, $2 E$

commentList the conjugacy classes of $A_{6}$ and determine their sizes. Hence prove that $A_{6}$ is simple.

Paper 2, Section II, 11E

commentProve that every finite integral domain is a field.

Let $F$ be a field and $f$ an irreducible polynomial in the polynomial ring $F[X]$. Prove that $F[X] /(f)$ is a field, where $(f)$ denotes the ideal generated by $f$.

Hence construct a field of 4 elements, and write down its multiplication table.

Construct a field of order 9 .

Paper 3, Section I, E

commentState and prove Hilbert's Basis Theorem.

Paper 3, Section II, E

commentLet $R$ be a ring, $M$ an $R$-module and $S=\left\{m_{1}, \ldots, m_{k}\right\}$ a subset of $M$. Define what it means to say $S$ spans $M$. Define what it means to say $S$ is an independent set.

We say $S$ is a basis for $M$ if $S$ spans $M$ and $S$ is an independent set. Prove that the following two statements are equivalent.

$S$ is a basis for $M$.

Every element of $M$ is uniquely expressible in the form $r_{1} m_{1}+\cdots+r_{k} m_{k}$ for some $r_{1}, \ldots, r_{k} \in R$.

We say $S$ generates $M$ freely if $S$ spans $M$ and any map $\Phi: S \rightarrow N$, where $N$ is an $R$-module, can be extended to an $R$-module homomorphism $\Theta: M \rightarrow N$. Prove that $S$ generates $M$ freely if and only if $S$ is a basis for $M$.

Let $M$ be an $R$-module. Are the following statements true or false? Give reasons.

(i) If $S$ spans $M$ then $S$ necessarily contains an independent spanning set for $M$.

(ii) If $S$ is an independent subset of $M$ then $S$ can always be extended to a basis for $M$.

Paper 4, Section I, E

commentLet $G$ be the abelian group generated by elements $a, b$ and $c$ subject to the relations: $3 a+6 b+3 c=0,9 b+9 c=0$ and $-3 a+3 b+6 c=0$. Express $G$ as a product of cyclic groups. Hence determine the number of elements of $G$ of order 3 .

Paper 4, Section II, E

comment(a) Consider the four following types of rings: Principal Ideal Domains, Integral Domains, Fields, and Unique Factorisation Domains. Arrange them in the form $A \Longrightarrow$ $B \Longrightarrow C \Longrightarrow D$ (where $A \Longrightarrow B$ means if a ring is of type $A$ then it is of type $B$ )

Prove that these implications hold. [You may assume that irreducibles in a Principal Ideal Domain are prime.] Provide examples, with brief justification, to show that these implications cannot be reversed.

(b) Let $R$ be a ring with ideals $I$ and $J$ satisfying $I \subseteq J$. Define $K$ to be the set $\{r \in R: r J \subseteq I\}$. Prove that $K$ is an ideal of $R$. If $J$ and $K$ are principal, prove that $I$ is principal.

Paper 1, Section II, G

comment(i) Consider the group $G=G L_{2}(\mathbb{R})$ of all 2 by 2 matrices with entries in $\mathbb{R}$ and non-zero determinant. Let $T$ be its subgroup consisting of all diagonal matrices, and $N$ be the normaliser of $T$ in $G$. Show that $N$ is generated by $T$ and $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$, and determine the quotient group $N / T$.

(ii) Now let $p$ be a prime number, and $F$ be the field of integers modulo $p$. Consider the group $G=G L_{2}(F)$ as above but with entries in $F$, and define $T$ and $N$ similarly. Find the order of the group $N$.

Paper 2, Section I, G

commentShow that every Euclidean domain is a PID. Define the notion of a Noetherian ring, and show that $\mathbb{Z}[i]$ is Noetherian by using the fact that it is a Euclidean domain.

Paper 2, Section II, G

comment(i) State the structure theorem for finitely generated modules over Euclidean domains.

(ii) Let $\mathbb{C}[X]$ be the polynomial ring over the complex numbers. Let $M$ be a $\mathbb{C}[X]$ module which is 4-dimensional as a $\mathbb{C}$-vector space and such that $(X-2)^{4} \cdot x=0$ for all $x \in M$. Find all possible forms we obtain when we write $M \cong \bigoplus_{i=1}^{m} \mathbb{C}[X] /\left(P_{i}^{n_{i}}\right)$ for irreducible $P_{i} \in \mathbb{C}[X]$ and $n_{i} \geqslant 1$.

(iii) Consider the quotient ring $M=\mathbb{C}[X] /\left(X^{3}+X\right)$ as a $\mathbb{C}[X]$-module. Show that $M$ is isomorphic as a $\mathbb{C}[X]$-module to the direct sum of three copies of $\mathbb{C}$. Give the isomorphism and its inverse explicitly.

Paper 3, Section I, $1 G$

commentDefine the notion of a free module over a ring. When $R$ is a PID, show that every ideal of $R$ is free as an $R$-module.

Paper 3, Section II, G

commentLet $R=\mathbb{C}[X, Y]$ be the polynomial ring in two variables over the complex numbers, and consider the principal ideal $I=\left(X^{3}-Y^{2}\right)$ of $R$.

(i) Using the fact that $R$ is a UFD, show that $I$ is a prime ideal of $R$. [Hint: Elements in $\mathbb{C}[X, Y]$ are polynomials in $Y$ with coefficients in $\mathbb{C}[X] .]$

(ii) Show that $I$ is not a maximal ideal of $R$, and that it is contained in infinitely many distinct proper ideals in $R$.

Paper 4, Section I, $2 G$

commentLet $p$ be a prime number, and $G$ be a non-trivial finite group whose order is a power of $p$. Show that the size of every conjugacy class in $G$ is a power of $p$. Deduce that the centre $Z$ of $G$ has order at least $p$.

Paper 4, Section II, 11G

commentLet $R$ be an integral domain, and $M$ be a finitely generated $R$-module.

(i) Let $S$ be a finite subset of $M$ which generates $M$ as an $R$-module. Let $T$ be a maximal linearly independent subset of $S$, and let $N$ be the $R$-submodule of $M$ generated by $T$. Show that there exists a non-zero $r \in R$ such that $r x \in N$ for every $x \in M$.

(ii) Now assume $M$ is torsion-free, i.e. $r x=0$ for $r \in R$ and $x \in M$ implies $r=0$ or $x=0$. By considering the map $M \rightarrow N$ mapping $x$ to $r x$ for $r$ as in (i), show that every torsion-free finitely generated $R$-module is isomorphic to an $R$-submodule of a finitely generated free $R$-module.

Paper 1, Section II, G

commentLet $G$ be a finite group. What is a Sylow $p$-subgroup of $G$ ?

Assuming that a Sylow $p$-subgroup $H$ exists, and that the number of conjugates of $H$ is congruent to $1 \bmod p$, prove that all Sylow $p$-subgroups are conjugate. If $n_{p}$ denotes the number of Sylow $p$-subgroups, deduce that

$n_{p} \equiv 1 \quad \bmod p \quad \text { and } \quad n_{p}|| G \mid$

If furthermore $G$ is simple prove that either $G=H$ or

$|G| \mid n_{p} \text { ! }$

Deduce that a group of order $1,000,000$ cannot be simple.

Paper 2, Section I, $2 G$

commentWhat does it mean to say that the finite group $G$ acts on the set $\Omega$ ?

By considering an action of the symmetry group of a regular tetrahedron on a set of pairs of edges, show there is a surjective homomorphism $S_{4} \rightarrow S_{3}$.

[You may assume that the symmetric group $S_{n}$ is generated by transpositions.]

Paper 2, Section II, G

commentState Gauss's Lemma. State Eisenstein's irreducibility criterion.

(i) By considering a suitable substitution, show that the polynomial $1+X^{3}+X^{6}$ is irreducible over $\mathbb{Q}$.

(ii) By working in $\mathbb{Z}_{2}[X]$, show that the polynomial $1-X^{2}+X^{5}$ is irreducible over $\mathbb{Q}$.

Paper 3, Section I, $1 \mathbf{G}$

commentWhat is a Euclidean domain?

Giving careful statements of any general results you use, show that in the ring $\mathbb{Z}[\sqrt{-3}], 2$ is irreducible but not prime.

Paper 3, Section II, G

commentFor each of the following assertions, provide either a proof or a counterexample as appropriate:

(i) The ring $\mathbb{Z}_{2}[X] /\left\langle X^{2}+X+1\right\rangle$ is a field.

(ii) The ring $\mathbb{Z}_{3}[X] /\left\langle X^{2}+X+1\right\rangle$ is a field.

(iii) If $F$ is a finite field, the ring $F[X]$ contains irreducible polynomials of arbitrarily large degree.

(iv) If $R$ is the ring $C[0,1]$ of continuous real-valued functions on the interval $[0,1]$, and the non-zero elements $f, g \in R$ satisfy $f \mid g$ and $g \mid f$, then there is some unit $u \in R$ with $f=u \cdot g$.

Paper 4, Section I, $2 G$

commentAn idempotent element of a ring $R$ is an element $e$ satisfying $e^{2}=e$. A nilpotent element is an element e satisfying $e^{N}=0$ for some $N \geqslant 0$.

Let $r \in R$ be non-zero. In the ring $R[X]$, can the polynomial $1+r X$ be (i) an idempotent, (ii) a nilpotent? Can $1+r X$ satisfy the equation $(1+r X)^{3}=(1+r X)$ ? Justify your answers.

Paper 4, Section II, G

commentLet $R$ be a commutative ring with unit 1. Prove that an $R$-module is finitely generated if and only if it is a quotient of a free module $R^{n}$, for some $n>0$.

Let $M$ be a finitely generated $R$-module. Suppose now $I$ is an ideal of $R$, and $\phi$ is an $R$-homomorphism from $M$ to $M$ with the property that

$\phi(M) \subset I \cdot M=\left\{m \in M \mid m=r m^{\prime} \quad \text { with } \quad r \in I, m^{\prime} \in M\right\}$

Prove that $\phi$ satisfies an equation

$\phi^{n}+a_{n-1} \phi^{n-1}+\cdots+a_{1} \phi+a_{0}=0$

where each $a_{j} \in I$. [You may assume that if $T$ is a matrix over $R$, then $\operatorname{adj}(T) T=$ $\operatorname{det} T$ (id), with id the identity matrix.]

Deduce that if $M$ satisfies $I \cdot M=M$, then there is some $a \in R$ satisfying

$a-1 \in I \quad \text { and } \quad a M=0 .$

Give an example of a finitely generated $\mathbb{Z}$-module $M$ and a proper ideal $I$ of $\mathbb{Z}$ satisfying the hypothesis $I \cdot M=M$, and for your example, give an explicit such element $a$.

Paper 1, Section II, F

comment(i) Suppose that $G$ is a finite group of order $p^{n} r$, where $p$ is prime and does not divide $r$. Prove the first Sylow theorem, that $G$ has at least one subgroup of order $p^{n}$, and state the remaining Sylow theorems without proof.

(ii) Suppose that $p, q$ are distinct primes. Show that there is no simple group of order $p q$.

Paper 2, Section I, F

commentShow that the quaternion group $Q_{8}=\{\pm 1, \pm i, \pm j, \pm k\}$, with $i j=k=-j i$, $i^{2}=j^{2}=k^{2}=-1$, is not isomorphic to the symmetry group $D_{8}$ of the square.

Paper 2, Section II, F

commentDefine the notion of a Euclidean domain and show that $\mathbb{Z}[i]$ is Euclidean.

Is $4+i$ prime in $\mathbb{Z}[i]$ ?

Paper 3, Section I, $1 F$

commentSuppose that $A$ is an integral domain containing a field $K$ and that $A$ is finitedimensional as a $K$-vector space. Prove that $A$ is a field.

Paper 3, Section II, F

commentSuppose that $A$ is a matrix over $\mathbb{Z}$. What does it mean to say that $A$ can be brought to Smith normal form?

Show that the structure theorem for finitely generated modules over $\mathbb{Z}$ (which you should state) follows from the existence of Smith normal forms for matrices over $\mathbb{Z}$.

Bring the matrix $\left(\begin{array}{cc}-4 & -6 \\ 2 & 2\end{array}\right)$ to Smith normal form.

Suppose that $M$ is the $\mathbb{Z}$-module with generators $e_{1}, e_{2}$, subject to the relations

$-4 e_{1}+2 e_{2}=-6 e_{1}+2 e_{2}=0$

Describe $M$ in terms of the structure theorem.

Paper 4, Section I, F

commentA ring $R$ satisfies the descending chain condition (DCC) on ideals if, for every sequence $I_{1} \supseteq I_{2} \supseteq I_{3} \supseteq \ldots$ of ideals in $R$, there exists $n$ with $I_{n}=I_{n+1}=I_{n+2}=\ldots$ Show that $\mathbb{Z}$ does not satisfy the DCC on ideals.

Paper 4, Section II, F

commentState and prove the Hilbert Basis Theorem.

Is every ring Noetherian? Justify your answer.

Paper 1, Section II, H

commentProve that the kernel of a group homomorphism $f: G \rightarrow H$ is a normal subgroup of the group $G$.

Show that the dihedral group $D_{8}$ of order 8 has a non-normal subgroup of order 2. Conclude that, for a group $G$, a normal subgroup of a normal subgroup of $G$ is not necessarily a normal subgroup of $G$.

Paper 2, Section I, $2 \mathrm{H}$

commentGive the definition of conjugacy classes in a group $G$. How many conjugacy classes are there in the symmetric group $S_{4}$ on four letters? Briefly justify your answer.

Paper 2, Section II, H

commentFor ideals $I, J$ of a ring $R$, their product $I J$ is defined as the ideal of $R$ generated by the elements of the form $x y$ where $x \in I$ and $y \in J$.

(1) Prove that, if a prime ideal $P$ of $R$ contains $I J$, then $P$ contains either $I$ or $J$.

(2) Give an example of $R, I$ and $J$ such that the two ideals $I J$ and $I \cap J$ are different from each other.

(3) Prove that there is a natural bijection between the prime ideals of $R / I J$ and the prime ideals of $R /(I \cap J)$.

Paper 3, Section I, H

commentLet $A$ be the ring of integers $\mathbb{Z}$ or the polynomial ring $\mathbb{C}[X]$. In each case, give an example of an ideal $I$ of $A$ such that the quotient ring $R=A / I$ has a non-trivial idempotent (an element $x \in R$ with $x \neq 0,1$ and $x^{2}=x$ ) and a non-trivial nilpotent element (an element $x \in R$ with $x \neq 0$ and $x^{n}=0$ for some positive integer $n$ ). Exhibit these elements and justify your answer.

Paper 3, Section II, H

commentLet $R$ be an integral domain and $R^{\times}$its group of units. An element of $S=R \backslash\left(R^{\times} \cup\{0\}\right)$ is irreducible if it is not a product of two elements in $S$. When $R$ is Noetherian, show that every element of $S$ is a product of finitely many irreducible elements of $S$.

Paper 4, Section I, H

commentLet $M$ be a free $\mathbb{Z}$-module generated by $e_{1}$ and $e_{2}$. Let $a, b$ be two non-zero integers, and $N$ be the submodule of $M$ generated by $a e_{1}+b e_{2}$. Prove that the quotient module $M / N$ is free if and only if $a, b$ are coprime.

Paper 4, Section II, $11 H$

commentLet $V=(\mathbb{Z} / 3 \mathbb{Z})^{2}$, a 2-dimensional vector space over the field $\mathbb{Z} / 3 \mathbb{Z}$, and let $e_{1}=\left(\begin{array}{c}1 \\ 0\end{array}\right), e_{2}=\left(\begin{array}{c}0 \\ 1\end{array}\right) \in V .$

(1) List all 1-dimensional subspaces of $V$ in terms of $e_{1}, e_{2}$. (For example, there is a subspace $\left\langle e_{1}\right\rangle$ generated by $\left.e_{1} .\right)$

(2) Consider the action of the matrix group

$G=G L_{2}(\mathbb{Z} / 3 \mathbb{Z})=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \mid a, b, c, d \in \mathbb{Z} / 3 \mathbb{Z}, \quad a d-b c \neq 0\right\}$

on the finite set $X$ of all 1-dimensional subspaces of $V$. Describe the stabiliser group $K$ of $\left\langle e_{1}\right\rangle \in X$. What is the order of $K$ ? What is the order of $G$ ?

(3) Let $H \subset G$ be the subgroup of all elements of $G$ which act trivially on $X$. Describe $H$, and prove that $G / H$ is isomorphic to $S_{4}$, the symmetric group on four letters.

Paper 1, Section II, F

commentProve that a principal ideal domain is a unique factorization domain.

Give, with justification, an example of an element of $\mathbb{Z}[\sqrt{-3}]$ which does not have a unique factorization as a product of irreducibles. Show how $\mathbb{Z}[\sqrt{-3}]$ may be embedded as a subring of index 2 in a ring $R$ (that is, such that the additive quotient group $R / \mathbb{Z}[\sqrt{-3}]$ has order 2) which is a principal ideal domain. [You should explain why $R$ is a principal ideal domain, but detailed proofs are not required.]

Paper 2, Section $\mathbf{I}$, $2 F$

commentState Sylow's theorems. Use them to show that a group of order 56 must have either a normal subgroup of order 7 or a normal subgroup of order 8 .

Paper 2, Section II, F

commentDefine the centre of a group, and prove that a group of prime-power order has a nontrivial centre. Show also that if the quotient group $G / Z(G)$ is cyclic, where $Z(G)$ is the centre of $G$, then it is trivial. Deduce that a non-abelian group of order $p^{3}$, where $p$ is prime, has centre of order $p$.

Let $F$ be the field of $p$ elements, and let $G$ be the group of $3 \times 3$ matrices over $F$ of the form

$\left(\begin{array}{lll} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array}\right)$

Identify the centre of $G$.

Paper 3, Section I, F

commentLet $F$ be a field. Show that the polynomial ring $F[X]$ is a principal ideal domain. Give, with justification, an example of an ideal in $F[X, Y]$ which is not principal.

Paper 3, Section II, F

commentLet $S$ be a multiplicatively closed subset of a ring $R$, and let $I$ be an ideal of $R$ which is maximal among ideals disjoint from $S$. Show that $I$ is prime.

If $R$ is an integral domain, explain briefly how one may construct a field $F$ together with an injective ring homomorphism $R \rightarrow F$.

Deduce that if $R$ is an arbitrary ring, $I$ an ideal of $R$, and $S$ a multiplicatively closed subset disjoint from $I$, then there exists a ring homomorphism $f: R \rightarrow F$, where $F$ is a field, such that $f(x)=0$ for all $x \in I$ and $f(y) \neq 0$ for all $y \in S$.

[You may assume that if $T$ is a multiplicatively closed subset of a ring, and $0 \notin T$, then there exists an ideal which is maximal among ideals disjoint from $T$.]

Paper 4, Section I, F

commentLet $M$ be a module over an integral domain $R$. An element $m \in M$ is said to be torsion if there exists a nonzero $r \in R$ with $\mathrm{rm}=0 ; M$ is said to be torsion-free if its only torsion element is 0 . Show that there exists a unique submodule $N$ of $M$ such that (a) all elements of $N$ are torsion and (b) the quotient module $M / N$ is torsion-free.

Paper 4, Section II, F

commentLet $R$ be a principal ideal domain. Prove that any submodule of a finitely-generated free module over $R$ is free.

An $R$-module $P$ is said to be projective if, whenever we have module homomorphisms $f: M \rightarrow N$ and $g: P \rightarrow N$ with $f$ surjective, there exists a homomorphism $h: P \rightarrow M$ with $f \circ h=g$. Show that any free module (over an arbitrary ring) is projective. Show also that a finitely-generated projective module over a principal ideal domain is free.

$3 . \mathrm{II} . 11 \mathrm{G}$

commentWhat is a Euclidean domain? Show that a Euclidean domain is a principal ideal domain.

Show that $\mathbb{Z}[\sqrt{-7}]$ is not a Euclidean domain (for any choice of norm), but that the ring

$\mathbb{Z}\left[\frac{1+\sqrt{-7}}{2}\right]$

is Euclidean for the norm function $N(z)=z \bar{z}$.

1.II.10G

comment(i) Show that $A_{4}$ is not simple.

(ii) Show that the group Rot $(D)$ of rotational symmetries of a regular dodecahedron is a simple group of order 60 .

(iii) Show that $\operatorname{Rot}(D)$ is isomorphic to $A_{5}$.

2.I.2G

commentWhat does it means to say that a complex number $\alpha$ is algebraic over $\mathbb{Q}$ ? Define the minimal polynomial of $\alpha$.

Suppose that $\alpha$ satisfies a nonconstant polynomial $f \in \mathbb{Z}[X]$ which is irreducible over $\mathbb{Z}$. Show that there is an isomorphism $\mathbb{Z}[X] /(f) \cong \mathbb{Z}[\alpha]$.

[You may assume standard results about unique factorisation, including Gauss's lemma.]

2.II.11G

commentLet $F$ be a field. Prove that every ideal of the ring $F\left[X_{1}, \ldots, X_{n}\right]$ is finitely generated.

Consider the set

$R=\left\{p(X, Y)=\sum c_{i j} X^{i} Y^{j} \in F[X, Y] \mid c_{0 j}=c_{j 0}=0 \text { whenever } j>0\right\}$

Show that $R$ is a subring of $F[X, Y]$ which is not Noetherian.

3.I.1G

commentLet $G$ be the abelian group generated by elements $a, b, c, d$ subject to the relations

$4 a-2 b+2 c+12 d=0, \quad-2 b+2 c=0, \quad 2 b+2 c=0, \quad 8 a+4 c+24 d=0$

Express $G$ as a product of cyclic groups, and find the number of elements of $G$ of order 2 .

4.I.2G

commentLet $n \geq 2$ be an integer. Show that the polynomial $\left(X^{n}-1\right) /(X-1)$ is irreducible over $\mathbb{Z}$ if and only if $n$ is prime.

[You may use Eisenstein's criterion without proof.]

4.II.11G

commentLet $R$ be a ring and $M$ an $R$-module. What does it mean to say that $M$ is a free $R$-module? Show that $M$ is free if there exists a submodule $N \subseteq M$ such that both $N$ and $M / N$ are free.

Let $M$ and $M^{\prime}$ be $R$-modules, and $N \subseteq M, N^{\prime} \subseteq M^{\prime}$ submodules. Suppose that $N \cong N^{\prime}$ and $M / N \cong M^{\prime} / N^{\prime}$. Determine (by proof or counterexample) which of the following statements holds:

(1) If $N$ is free then $M \cong M^{\prime}$.

(2) If $M / N$ is free then $M \cong M^{\prime}$.

1.II.10G

comment(i) State a structure theorem for finitely generated abelian groups.

(ii) If $K$ is a field and $f$ a polynomial of degree $n$ in one variable over $K$, what is the maximal number of zeroes of $f$ ? Justify your answer in terms of unique factorization in some polynomial ring, or otherwise.

(iii) Show that any finite subgroup of the multiplicative group of non-zero elements of a field is cyclic. Is this true if the subgroup is allowed to be infinite?

2.I.2G

commentDefine the term Euclidean domain.

Show that the ring of integers $\mathbb{Z}$ is a Euclidean domain.

2.II.11G

comment(i) Give an example of a Noetherian ring and of a ring that is not Noetherian. Justify your answers.

(ii) State and prove Hilbert's basis theorem.

3.I.1G

commentWhat are the orders of the groups $G L_{2}\left(\mathbb{F}_{p}\right)$ and $S L_{2}\left(\mathbb{F}_{p}\right)$ where $\mathbb{F}_{p}$ is the field of $p$ elements?

3.II.11G

comment(i) State the Sylow theorems for Sylow $p$-subgroups of a finite group.

(ii) Write down one Sylow 3-subgroup of the symmetric group $S_{5}$ on 5 letters. Calculate the number of Sylow 3-subgroups of $S_{5}$.

4.I.2G

commentIf $p$ is a prime, how many abelian groups of order $p^{4}$ are there, up to isomorphism?

4.II.11G

commentA regular icosahedron has 20 faces, 12 vertices and 30 edges. The group $G$ of its rotations acts transitively on the set of faces, on the set of vertices and on the set of edges.

(i) List the conjugacy classes in $G$ and give the size of each.

(ii) Find the order of $G$ and list its normal subgroups.

[A normal subgroup of $G$ is a union of conjugacy classes in $G$.]

1.II.10E

commentFind all subgroups of indices $2,3,4$ and 5 in the alternating group $A_{5}$ on 5 letters. You may use any general result that you choose, provided that you state it clearly, but you must justify your answers.

[You may take for granted the fact that $A_{4}$ has no subgroup of index 2.]

2.I.2E

comment(i) Give the definition of a Euclidean domain and, with justification, an example of a Euclidean domain that is not a field.

(ii) State the structure theorem for finitely generated modules over a Euclidean domain.

(iii) In terms of your answer to (ii), describe the structure of the $\mathbb{Z}$-module $M$ with generators $\left\{m_{1}, m_{2}, m_{3}\right\}$ and relations $2 m_{3}=2 m_{2}, 4 m_{2}=0$.

2.II.11E

comment(i) Prove the first Sylow theorem, that a finite group of order $p^{n} r$ with $p$ prime and $p$ not dividing the integer $r$ has a subgroup of order $p^{n}$.

(ii) State the remaining Sylow theorems.

(iii) Show that if $p$ and $q$ are distinct primes then no group of order $p q$ is simple.

3.I.1E

comment(i) Give an example of an integral domain that is not a unique factorization domain.

(ii) For which integers $n$ is $\mathbb{Z} / n \mathbb{Z}$ an integral domain?

3.II.11E

commentSuppose that $R$ is a ring. Prove that $R[X]$ is Noetherian if and only if $R$ is Noetherian.

4.I $. 2 \mathrm{E} \quad$

commentHow many elements does the ring $\mathbb{Z}[X] /\left(3, X^{2}+X+1\right)$ have?

Is this ring an integral domain?

Briefly justify your answers.

4.II.11E

comment(a) Suppose that $R$ is a commutative ring, $M$ an $R$-module generated by $m_{1}, \ldots, m_{n}$ and $\phi \in \operatorname{End}_{R}(M)$. Show that, if $A=\left(a_{i j}\right)$ is an $n \times n$ matrix with entries in $R$ that represents $\phi$ with respect to this generating set, then in the sub-ring $R[\phi]$ of $\operatorname{End}_{R}(M)$ we have $\operatorname{det}\left(a_{i j}-\phi \delta_{i j}\right)=0 .$

[Hint: $A$ is a matrix such that $\phi\left(m_{i}\right)=\sum a_{i j} m_{j}$ with $a_{i j} \in R$. Consider the matrix $C=\left(a_{i j}-\phi \delta_{i j}\right)$ with entries in $R[\phi]$ and use the fact that for any $n \times n$ matrix $N$ over any commutative ring, there is a matrix $N^{\prime}$ such that $N^{\prime} N=(\operatorname{det} N) 1_{n}$.]

(b) Suppose that $k$ is a field, $V$ a finite-dimensional $k$-vector space and that $\phi \in \operatorname{End}_{k}(V)$. Show that if $A$ is the matrix of $\phi$ with respect to some basis of $V$ then $\phi$ satisfies the characteristic equation $\operatorname{det}(A-\lambda 1)=0$ of $A$.

$3 . \mathrm{II} . 11 \mathrm{C}$

comment(i) Define a primitive polynomial in $\mathbb{Z}[x]$, and prove that the product of two primitive polynomials is primitive. Deduce that $\mathbb{Z}[x]$ is a unique factorization domain.

(ii) Prove that

$\mathbb{Q}[x] /\left(x^{5}-4 x+2\right)$

is a field. Show, on the other hand, that

$\mathbb{Z}[x] /\left(x^{5}-4 x+2\right)$

is an integral domain, but is not a field.

1.II.10C

commentLet $G$ be a group, and $H$ a subgroup of finite index. By considering an appropriate action of $G$ on the set of left cosets of $H$, prove that $H$ always contains a normal subgroup $K$ of $G$ such that the index of $K$ in $G$ is finite and divides $n$ !, where $n$ is the index of $H$ in $G$.

Now assume that $G$ is a finite group of order $p q$, where $p$ and $q$ are prime numbers with $p<q$. Prove that the subgroup of $G$ generated by any element of order $q$ is necessarily normal.

2.I.2C

commentDefine an automorphism of a group $G$, and the natural group law on the set $\operatorname{Aut}(G)$ of all automorphisms of $G$. For each fixed $h$ in $G$, put $\psi(h)(g)=h g h^{-1}$ for all $g$ in $G$. Prove that $\psi(h)$ is an automorphism of $G$, and that $\psi$ defines a homomorphism from $G$ into $\operatorname{Aut}(G)$.

2.II.11C

commentLet $A$ be the abelian group generated by two elements $x, y$, subject to the relation $6 x+9 y=0$. Give a rigorous explanation of this statement by defining $A$ as an appropriate quotient of a free abelian group of rank 2. Prove that $A$ itself is not a free abelian group, and determine the exact structure of $A$.

3.I.1C

commentDefine what is meant by two elements of a group $G$ being conjugate, and prove that this defines an equivalence relation on $G$. If $G$ is finite, sketch the proof that the cardinality of each conjugacy class divides the order of $G$.

4.I.2C

commentState Eisenstein's irreducibility criterion. Let $n$ be an integer $>1$. Prove that $1+x+\ldots+x^{n-1}$ is irreducible in $\mathbb{Z}[x]$ if and only if $n$ is a prime number.

4.II.11C

commentLet $R$ be the ring of Gaussian integers $\mathbb{Z}[i]$, where $i^{2}=-1$, which you may assume to be a unique factorization domain. Prove that every prime element of $R$ divides precisely one positive prime number in $\mathbb{Z}$. List, without proof, the prime elements of $R$, up to associates.

Let $p$ be a prime number in $\mathbb{Z}$. Prove that $R / p R$ has cardinality $p^{2}$. Prove that $R / 2 R$ is not a field. If $p \equiv 3 \bmod 4$, show that $R / p R$ is a field. If $p \equiv 1 \bmod 4$, decide whether $R / p R$ is a field or not, justifying your answer.

$1 . \mathrm{I} . 2 \mathrm{~F} \quad$

commentLet $G$ be a finite group of order $n$. Let $H$ be a subgroup of $G$. Define the normalizer $N(H)$ of $H$, and prove that the number of distinct conjugates of $H$ is equal to the index of $N(H)$ in $G$. If $p$ is a prime dividing $n$, deduce that the number of Sylow $p$-subgroups of $G$ must divide $n$.

[You may assume the existence and conjugacy of Sylow subgroups.]

Prove that any group of order 72 must have either 1 or 4 Sylow 3-subgroups.

$3 . \mathrm{II} . 14 \mathrm{~F} \quad$

commentLet $L$ be the group $\mathbb{Z}^{3}$ consisting of 3-dimensional row vectors with integer components. Let $M$ be the subgroup of $L$ generated by the three vectors

$u=(1,2,3), v=(2,3,1), w=(3,1,2) \text {. }$

(i) What is the index of $M$ in $L$ ?

(ii) Prove that $M$ is not a direct summand of $L$.

(iii) Is the subgroup $N$ generated by $u$ and $v$ a direct summand of $L$ ?

(iv) What is the structure of the quotient group $L / M$ ?

$3 . \mathrm{I} . 2 \mathrm{~F} \quad$

Let $R$ be the subring of all $z$ in $\mathbb{C}$ of the form

$z=\frac{a+b \sqrt{-3}}{2}$

where $a$ and $b$ are in $\mathbb{Z}$ and $a \equiv b(\bmod 2)$. Prove that $N(z)=z \bar{z}$ is a non-negative element of $\mathbb{Z}$, for all $z$ in $R$. Prove that the multiplicative group of units of $R$