• # Paper 3 , Section II, D

Let $S(X)$ denote the group of permutations of a finite set $X$. Show that every permutation $\sigma \in S(X)$ can be written as a product of disjoint cycles. Explain briefly why two permutations in $S(X)$ are conjugate if and only if, when they are written as the product of disjoint cycles, they have the same number of cycles of length $n$ for each possible value of $n$.

Let $\ell(\sigma)$ denote the number of disjoint cycles, including 1-cycles, required when $\sigma$ is written as a product of disjoint cycles. Let $\tau$ be a transposition in $S(X)$ and $\sigma$ any permutation in $S(X)$. Prove that $\ell(\tau \sigma)=\ell(\sigma) \pm 1$.

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• # Paper 3, Section I, D

State and prove Lagrange's theorem. Give an example to show that an integer $k$ may divide the order of a group $G$ without there being a subgroup of order $k$.

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• # Paper 3, Section I, D

Show that every orthogonal $2 \times 2$ matrix $R$ is the product of at most two reflections in lines through the origin.

Every isometry of the Euclidean plane $\mathbb{R}^{2}$ can be written as the composition of an orthogonal matrix and a translation. Deduce from this that every isometry of the Euclidean plane $\mathbb{R}^{2}$ is a product of reflections.

Give an example of an isometry of $\mathbb{R}^{2}$ that is not the product of fewer than three reflections. Justify your answer.

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• # Paper 3, Section II, D

What does it mean to say that a subgroup $K$ of a group $G$ is normal?

Let $\phi: G \rightarrow H$ be a group homomorphism. Is the kernel of $\phi$ always a subgroup of $G$ ? Is it always a normal subgroup? Is the image of $\phi$ always a subgroup of $H$ ? Is it always a normal subgroup? Justify your answers.

Let $\mathrm{SL}(2, \mathbb{Z})$ denote the set of $2 \times 2$ matrices $\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ with $a, b, c, d \in \mathbb{Z}$ and $a d-b c=1$. Show that $\mathrm{SL}(2, \mathbb{Z})$ is a group under matrix multiplication. Similarly, when $\mathbb{Z}_{2}$ denotes the integers modulo 2 , let $\mathrm{SL}\left(2, \mathbb{Z}_{2}\right)$ denote the set of $2 \times 2$ matrices $\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ with $a, b, c, d \in \mathbb{Z}_{2}$ and $a d-b c=1$. Show that $\mathrm{SL}\left(2, \mathbb{Z}_{2}\right)$ is also a group under matrix multiplication.

Let $f: \mathbb{Z} \rightarrow \mathbb{Z}_{2}$ send each integer to its residue modulo 2 . Show that

$\phi: \mathrm{SL}(2, \mathbb{Z}) \rightarrow \mathrm{SL}\left(2, \mathbb{Z}_{2}\right) ; \quad\left(\begin{array}{cc} a & b \\ c & d \end{array}\right) \mapsto\left(\begin{array}{ll} f(a) & f(b) \\ f(c) & f(d) \end{array}\right)$

is a group homomorphism. Show that the image of $\phi$ is isomorphic to a permutation group.

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• # Paper 3, Section II, D

Define the cross-ratio $\left[a_{0}, a_{1}, a_{2}, z\right]$ of four points $a_{0}, a_{1}, a_{2}, z$ in $\mathbb{C} \cup\{\infty\}$, with $a_{0}, a_{1}, a_{2}$ distinct.

Let $a_{0}, a_{1}, a_{2}$ be three distinct points. Show that, for every value $w \in \mathbb{C} \cup\{\infty\}$, there is a unique point $z \in \mathbb{C} \cup\{\infty\}$ with $\left[a_{0}, a_{1}, a_{2}, z\right]=w$. Let $S$ be the set of points $z$ for which the cross-ratio $\left[a_{0}, a_{1}, a_{2}, z\right]$ is in $\mathbb{R} \cup\{\infty\}$. Show that $S$ is either a circle or else a straight line together with $\infty$.

A map $J: \mathbb{C} \cup\{\infty\} \rightarrow \mathbb{C} \cup\{\infty\}$ satisfies

$\left[a_{0}, a_{1}, a_{2}, J(z)\right]=\overline{\left[a_{0}, a_{1}, a_{2}, z\right]}$

for each value of $z$. Show that this gives a well-defined map $J$ with $J^{2}$ equal to the identity.

When the three points $a_{0}, a_{1}, a_{2}$ all lie on the real line, show that $J$ must be the conjugation map $J: z \mapsto \bar{z}$. Deduce from this that, for any three distinct points $a_{0}, a_{1}, a_{2}$, the map $J$ depends only on the circle (or straight line) through $a_{0}, a_{1}, a_{2}$ and not on their particular values.

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• # Paper 3, Section II, D

State and prove the orbit-stabilizer theorem.

Let $G$ be the group of all symmetries of a regular octahedron, including both orientation-preserving and orientation-reversing symmetries. How many symmetries are there in the group $G$ ? Let $D$ be the set of straight lines that join a vertex of the octahedron to the opposite vertex. How many lines are there in the set $D$ ? Identify the stabilizer in $G$ of one of the lines in $D$.

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• # Paper 3, Section $I$, B

State the value of $\partial x_{i} / \partial x_{j}$ and find $\partial r / \partial x_{j}$, where $r=|\mathbf{x}|$.

A vector field $\mathbf{u}$ is given by

$\mathbf{u}=\frac{\mathbf{k}}{r}+\frac{(\mathbf{k} \cdot \mathbf{x}) \mathbf{x}}{r^{3}}$

where $\mathbf{k}$ is a constant vector. Calculate the second-rank tensor $d_{i j}=\partial u_{i} / \partial x_{j}$ using suffix notation, and show that $d_{i j}$ splits naturally into symmetric and antisymmetric parts. Deduce that $\boldsymbol{\nabla} \cdot \mathbf{u}=0$ and that

$\boldsymbol{\nabla} \times \mathbf{u}=\frac{2 \mathbf{k} \times \mathbf{x}}{r^{3}}$

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• # Paper 3, Section I, B

What does it mean for a vector field $\mathbf{F}$ to be irrotational ?

The field $\mathbf{F}$ is irrotational and $\mathbf{x}_{0}$ is a given point. Write down a scalar potential $V(\mathbf{x})$ with $\mathbf{F}=-\nabla V$ and $V\left(\mathbf{x}_{0}\right)=0$. Show that this potential is well defined.

For what value of $m$ is the field $\frac{\cos \theta \cos \phi}{r} \mathbf{e}_{\theta}+\frac{m \sin \phi}{r} \mathbf{e}_{\phi}$ irrotational, where $(r, \theta, \phi)$ are spherical polar coordinates? What is the corresponding potential $V(\mathbf{x})$ when $\mathbf{x}_{0}$ is the point $r=1, \theta=0$ ?

$\left[\text { In spherical polar coordinates } \nabla \times \mathbf{F}=\frac{1}{r^{2} \sin \theta} \mid \begin{array}{ccc} \mathbf{e}_{r} & r \mathbf{e}_{\theta} & r \sin \theta \mathbf{e}_{\phi} \\ \partial / \partial r & \partial / \partial \theta & \partial / \partial \phi \\ F_{r} & r F_{\theta} & r \sin \theta F_{\phi} \end{array}\right] .$

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• # Paper 3, Section II, B

State the divergence theorem for a vector field $\mathbf{u}(\mathbf{x})$ in a region $V$ of $\mathbb{R}^{3}$ bounded by a smooth surface $S$.

Let $f(x, y, z)$ be a homogeneous function of degree $n$, that is, $f(k x, k y, k z)=$ $k^{n} f(x, y, z)$ for any real number $k$. By differentiating with respect to $k$, show that

$\mathbf{x} \cdot \nabla f=n f$

Deduce that

$\tag{†} \int_{V} f \mathrm{~d} V=\frac{1}{n+3} \int_{S} f \mathbf{x} \cdot \mathbf{d} \mathbf{A}$

Let $V$ be the cone $0 \leqslant z \leqslant \alpha, \alpha \sqrt{x^{2}+y^{2}} \leqslant z$, where $\alpha$ is a positive constant. Verify that $(†)$ holds for the case $f=z^{4}+\alpha^{4}\left(x^{2}+y^{2}\right)^{2}$.

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• # Paper 3, Section II, B

A second-rank tensor $T(\mathbf{y})$ is defined by

$T_{i j}(\mathbf{y})=\int_{S}\left(y_{i}-x_{i}\right)\left(y_{j}-x_{j}\right)|\mathbf{y}-\mathbf{x}|^{2 n-2} \mathrm{~d} A(\mathbf{x})$

where $\mathbf{y}$ is a fixed vector with $|\mathbf{y}|=a, n>-1$, and the integration is over all points $\mathbf{x}$ lying on the surface $S$ of the sphere of radius $a$, centred on the origin. Explain briefly why $T$ might be expected to have the form

$T_{i j}=\alpha \delta_{i j}+\beta y_{i} y_{j}$

where $\alpha$ and $\beta$ are scalar constants.

Show that $\mathbf{y} \cdot(\mathbf{y}-\mathbf{x})=a^{2}(1-\cos \theta)$, where $\theta$ is the angle between $\mathbf{y}$ and $\mathbf{x}$, and find a similar expression for $|\mathbf{y}-\mathbf{x}|^{2}$. Using suitably chosen spherical polar coordinates, show that

$y_{i} T_{i j} y_{j}=\frac{\pi a^{2}(2 a)^{2 n+2}}{n+2}$

Hence, by evaluating another scalar integral, determine $\alpha$ and $\beta$, and find the value of $n$ for which $T$ is isotropic.

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• # Paper 3, Section II, B

Give a necessary condition for a given vector field $\mathbf{J}$ to be the curl of another vector field $\mathbf{B}$. Is the vector field $\mathbf{B}$ unique? If not, explain why not.

State Stokes' theorem and use it to evaluate the area integral

$\int_{S}\left(y^{2}, z^{2}, x^{2}\right) \cdot \mathbf{d} \mathbf{A}$

where $S$ is the half of the ellipsoid

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$

that lies in $z \geqslant 0$, and the area element dA points out of the ellipsoid.

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• # Paper 3, Section II, B

Let $S$ be a bounded region of $\mathbb{R}^{2}$ and $\partial S$ be its boundary. Let $u$ be the unique solution to Laplace's equation in $S$, subject to the boundary condition $u=f$ on $\partial S$, where $f$ is a specified function. Let $w$ be any smooth function with $w=f$ on $\partial S$. By writing $w=u+\delta$, or otherwise, show that

$\int_{S}|\nabla w|^{2} \mathrm{~d} A \geqslant \int_{S}|\nabla u|^{2} \mathrm{~d} A$

Let $S$ be the unit disc in $\mathbb{R}^{2}$. By considering functions of the form $g(r) \cos \theta$ on both sides of $(*)$, where $r$ and $\theta$ are polar coordinates, deduce that

$\int_{0}^{1}\left(r\left(\frac{\mathrm{d} g}{\mathrm{~d} r}\right)^{2}+\frac{g^{2}}{r}\right) \mathrm{d} r \geqslant 1$

for any differentiable function $g(r)$ satisfying $g(1)=1$ and for which the integral converges at $r=0$.

$\left[\nabla f(r, \theta)=\left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}\right), \quad \nabla^{2} f(r, \theta)=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial f}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} f}{\partial \theta^{2}} \cdot\right]$

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