Part IA, 2011, Paper 3

Groups

• Paper 3, Section I, D

  Groups | Part IA, 2011

  State and prove Lagrange's Theorem.

  Show that the dihedral group of order $2 n$ has a subgroup of order $k$ for every $k$ dividing $2 n$.

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• Paper 3, Section I, D

  Groups | Part IA, 2011

  (a) Let $G$ be the group of symmetries of the cube, and consider the action of $G$ on the set of edges of the cube. Determine the stabilizer of an edge and its orbit. Hence compute the order of $G$.

  (b) The symmetric group $S_{n}$ acts on the set $X=\{1, \ldots, n\}$, and hence acts on $X \times X$ by $g(x, y)=(g x, g y)$. Determine the orbits of $S_{n}$ on $X \times X$.

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• Paper 3, Section II, $7 \mathrm{D}$

  Groups | Part IA, 2011

  (a) State the orbit-stabilizer theorem.

  Let a group $G$ act on itself by conjugation. Define the centre $Z(G)$ of $G$, and show that $Z(G)$ consists of the orbits of size 1 . Show that $Z(G)$ is a normal subgroup of $G$.

  (b) Now let $|G|=p^{n}$, where $p$ is a prime and $n \geqslant 1$. Show that if $G$ acts on a set $X$, and $Y$ is an orbit of this action, then either $|Y|=1$ or $p$ divides $|Y|$.

  Show that $|Z(G)|>1$.

  By considering the set of elements of $G$ that commute with a fixed element $x$ not in $Z(G)$, show that $Z(G)$ cannot have order $p^{n-1}$.

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• Paper 3, Section II, D

  Groups | Part IA, 2011

  (a) Let $G$ be a finite group and let $H$ be a subgroup of $G$. Show that if $|G|=2|H|$ then $H$ is normal in $G$.

  Show that the dihedral group $D_{2 n}$ of order $2 n$ has a normal subgroup different from both $D_{2 n}$ and $\{e\}$.

  For each integer $k \geqslant 3$, give an example of a finite group $G$, and a subgroup $H$, such that $|G|=k|H|$ and $H$ is not normal in $G$.

  (b) Show that $A_{5}$ is a simple group.

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• Paper 3, Section II, D

  Groups | Part IA, 2011

  (a) Let

  $$S L_{2}(\mathbb{Z})=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \mid a d-b c=1, \quad a, b, c, d \in \mathbb{Z}\right\}$$

  and, for a prime $p$, let

  $$S L_{2}\left(\mathbb{F}_{p}\right)=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \mid a d-b c=1, \quad a, b, c, d \in \mathbb{F}_{p}\right\}$$

  where $\mathbb{F}_{p}$ consists of the elements $0,1, \ldots, p-1$, with addition and multiplication mod $p$.

  Show that $S L_{2}(\mathbb{Z})$ and $S L_{2}\left(\mathbb{F}_{p}\right)$ are groups under matrix multiplication.

  [You may assume that matrix multiplication is associative, and that the determinant of a product equals the product of the determinants.]

  By defining a suitable homomorphism from $S L_{2}(\mathbb{Z}) \rightarrow S L_{2}\left(\mathbb{F}_{5}\right)$, show that

  $$\left\{\left(\begin{array}{cc} 1+5 a & 5 b \\ 5 c & 1+5 d \end{array}\right) \in S L_{2}(\mathbb{Z}) \mid a, b, c, d \in \mathbb{Z}\right\}$$

  is a normal subgroup of $S L_{2}(\mathbb{Z})$.

  (b) Define the group $G L_{2}\left(\mathbb{F}_{5}\right)$, and show that it has order 480 . By defining a suitable homomorphism from $G L_{2}\left(\mathbb{F}_{5}\right)$ to another group, which should be specified, show that the order of $S L_{2}\left(\mathbb{F}_{5}\right)$ is 120 .

  Find a subgroup of $G L_{2}\left(\mathbb{F}_{5}\right)$ of index 2 .

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• Paper 3, Section II, D

  Groups | Part IA, 2011

  (a) Let $G$ be a finite group, and let $g \in G$. Define the order of $g$ and show it is finite. Show that if $g$ is conjugate to $h$, then $g$ and $h$ have the same order.

  (b) Show that every $g \in S_{n}$ can be written as a product of disjoint cycles. For $g \in S_{n}$, describe the order of $g$ in terms of the cycle decomposition of $g$.

  (c) Define the alternating group $A_{n}$. What is the condition on the cycle decomposition of $g \in S_{n}$ that characterises when $g \in A_{n}$ ?

  (d) Show that, for every $n, A_{n+2}$ has a subgroup isomorphic to $S_{n}$.

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Vector Calculus

• Paper 3, Section I, C

  Vector Calculus | Part IA, 2011

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

  Vector fields $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^{3}$ are given by $\mathbf{u}=r^{\alpha} \mathbf{x}$ and $\mathbf{v}=\mathbf{k} \times \mathbf{u}$, where $\alpha$ is a constant and $\mathbf{k}$ is a constant vector. Calculate the second-rank tensor $d_{i j}=\partial u_{i} / \partial x_{j}$, and deduce that $\boldsymbol{\nabla} \times \mathbf{u}=\mathbf{0}$ and $\boldsymbol{\nabla} \cdot \mathbf{v}=0$. When $\alpha=-3$, show that $\boldsymbol{\nabla} \cdot \mathbf{u}=0$ and

  $$\nabla \times \mathbf{v}=\frac{3(\mathbf{k} \cdot \mathbf{x}) \mathbf{x}-\mathbf{k} r^{2}}{r^{5}}$$

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• Paper 3, Section I, C

  Vector Calculus | Part IA, 2011

  Cartesian coordinates $x, y, z$ and spherical polar coordinates $r, \theta, \phi$ are related by

  $$x=r \sin \theta \cos \phi, \quad y=r \sin \theta \sin \phi, \quad z=r \cos \theta$$

  Find scalars $h_{r}, h_{\theta}, h_{\phi}$ and unit vectors $\mathbf{e}_{r}, \mathbf{e}_{\theta}, \mathbf{e}_{\phi}$ such that

  $$\mathrm{d} \mathbf{x}=h_{r} \mathbf{e}_{r} \mathrm{~d} r+h_{\theta} \mathbf{e}_{\theta} \mathrm{d} \theta+h_{\phi} \mathbf{e}_{\phi} \mathrm{d} \phi$$

  Verify that the unit vectors are mutually orthogonal.

  Hence calculate the area of the open surface defined by $\theta=\alpha, 0 \leqslant r \leqslant R$, $0 \leqslant \phi \leqslant 2 \pi$, where $\alpha$ and $R$ are constants.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2011

  The vector fields $\mathbf{A}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ obey the evolution equations

  $$\begin{aligned} \frac{\partial \mathbf{A}}{\partial t} &=\mathbf{u} \times(\boldsymbol{\nabla} \times \mathbf{A})+\nabla \psi \\ \frac{\partial \mathbf{B}}{\partial t} &=(\mathbf{B} \cdot \nabla) \mathbf{u}-(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{B} \end{aligned}$$

  where $\mathbf{u}$ is a given vector field and $\psi$ is a given scalar field. Use suffix notation to show that the scalar field $h=\mathbf{A} \cdot \mathbf{B}$ obeys an evolution equation of the form

  $$\frac{\partial h}{\partial t}=\mathbf{B} \cdot \nabla f-\mathbf{u} \cdot \nabla h$$

  where the scalar field $f$ should be identified.

  Suppose that $\boldsymbol{\nabla} \cdot \mathbf{B}=0$ and $\boldsymbol{\nabla} \cdot \mathbf{u}=0$. Show that, if $\mathbf{u} \cdot \mathbf{n}=\mathbf{B} \cdot \mathbf{n}=0$ on the surface $S$ of a fixed volume $V$ with outward normal $\mathbf{n}$, then

  $$\frac{\mathrm{d} H}{\mathrm{~d} t}=0, \quad \text { where } H=\int_{V} h \mathrm{~d} V .$$

  Suppose that $\mathbf{A}=a r^{2} \sin \theta \mathbf{e}_{\theta}+r\left(a^{2}-r^{2}\right) \sin \theta \mathbf{e}_{\phi}$ with respect to spherical polar coordinates, and that $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. Show that

  $$h=a r^{2}\left(a^{2}+r^{2}\right) \sin ^{2} \theta$$

  and calculate the value of $H$ when $V$ is the sphere $r \leqslant a$.

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

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2011

  The electric field $\mathbf{E}(\mathbf{x})$ due to a static charge distribution with density $\rho(\mathbf{x})$ satisfies

  $$\mathbf{E}=-\boldsymbol{\nabla} \phi, \quad \boldsymbol{\nabla} \cdot \mathbf{E}=\frac{\rho}{\varepsilon_{0}},$$

  where $\phi(\mathbf{x})$ is the corresponding electrostatic potential and $\varepsilon_{0}$ is a constant.

  (a) Show that the total charge $Q$ contained within a closed surface $S$ is given by Gauss' Law

  $$Q=\varepsilon_{0} \int_{S} \mathbf{E} \cdot \mathrm{d} \mathbf{S} .$$

  Assuming spherical symmetry, deduce the electric field and potential due to a point charge $q$ at the origin i.e. for $\rho(\mathbf{x})=q \delta(\mathbf{x})$.

  (b) Let $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$, with potentials $\phi_{1}$ and $\phi_{2}$ respectively, be the solutions to (1) arising from two different charge distributions with densities $\rho_{1}$ and $\rho_{2}$. Show that

  $$\frac{1}{\varepsilon_{0}} \int_{V} \phi_{1} \rho_{2} \mathrm{~d} V+\int_{\partial V} \phi_{1} \nabla \phi_{2} \cdot \mathrm{d} \mathbf{S}=\frac{1}{\varepsilon_{0}} \int_{V} \phi_{2} \rho_{1} \mathrm{~d} V+\int_{\partial V} \phi_{2} \nabla \phi_{1} \cdot \mathrm{d} \mathbf{S}$$

  for any region $V$ with boundary $\partial V$, where $\mathrm{d} \mathbf{S}$ points out of $V$.

  (c) Suppose that $\rho_{1}(\mathbf{x})=0$ for $|\mathbf{x}| \leqslant a$ and that $\phi_{1}(\mathbf{x})=\Phi$, a constant, on $|\mathbf{x}|=a$. Use the results of (a) and (b) to show that

  $$\Phi=\frac{1}{4 \pi \varepsilon_{0}} \int_{r>a} \frac{\rho_{1}(\mathbf{x})}{r} \mathrm{~d} V$$

  [You may assume that $\phi_{1} \rightarrow 0$ as $|\mathbf{x}| \rightarrow \infty$ sufficiently rapidly that any integrals over the 'sphere at infinity' in (2) are zero.]

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2011

  State the divergence theorem for a vector field $\mathbf{u}(\mathbf{x})$ in a region $V$ bounded by a piecewise smooth surface $S$ with outward normal $\mathbf{n}$.

  Show, by suitable choice of $\mathbf{u}$, that

  $$\int_{V} \nabla f \mathrm{~d} V=\int_{S} f \mathrm{~d} \mathbf{S}$$

  for a scalar field $f(\mathbf{x})$.

  Let $V$ be the paraboloidal region given by $z \geqslant 0$ and $x^{2}+y^{2}+c z \leqslant a^{2}$, where $a$ and $c$ are positive constants. Verify that $(*)$ holds for the scalar field $f=x z$.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2011

  Write down the most general isotropic tensors of rank 2 and 3. Use the tensor transformation law to show that they are, indeed, isotropic.

  Let $V$ be the sphere $0 \leqslant r \leqslant a$. Explain briefly why

  $$T_{i_{1} \ldots i_{n}}=\int_{V} x_{i_{1}} \ldots x_{i_{n}} \mathrm{~d} V$$

  is an isotropic tensor for any $n$. Hence show that

  $$\int_{V} x_{i} x_{j} \mathrm{~d} V=\alpha \delta_{i j}, \quad \int_{V} x_{i} x_{j} x_{k} \mathrm{~d} V=0 \quad \text { and } \int_{V} x_{i} x_{j} x_{k} x_{l} \mathrm{~d} V=\beta\left(\delta_{i j} \delta_{k l}+\delta_{i k} \delta_{j l}+\delta_{i l} \delta_{j k}\right)$$

  for some scalars $\alpha$ and $\beta$, which should be determined using suitable contractions of the indices or otherwise. Deduce the value of

  $$\int_{V} \mathbf{x} \times(\boldsymbol{\Omega} \times \mathbf{x}) \mathrm{d} V$$

  where $\boldsymbol{\Omega}$ is a constant vector.

  [You may assume that the most general isotropic tensor of rank 4 is

  $$\lambda \delta_{i j} \delta_{k l}+\mu \delta_{i k} \delta_{j l}+\nu \delta_{i l} \delta_{j k}$$

  where $\lambda, \mu$ and $\nu$ are scalars.]

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