Part IA, 2021, Paper 3

Groups

• Paper 3 , Section I, D

  Groups | Part IA, 2021

  Let $G$ be a finite group and denote the centre of $G$ by $Z(G)$. Prove that if the quotient group $G / Z(G)$ is cyclic then $G$ is abelian. Does there exist a group $H$ such that (i) $|H / Z(H)|=7$ ? (ii) $|H| Z(H) \mid=6$ ?

  Justify your answers.

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• Paper 3, Section I, D

  Groups | Part IA, 2021

  Let $g$ and $h$ be elements of a group $G$. What does it mean to say $g$ and $h$ are conjugate in $G$ ? Prove that if two elements in a group are conjugate then they have the same order.

  Define the Möbius group $\mathcal{M}$. Prove that if $g, h \in \mathcal{M}$ are conjugate they have the same number of fixed points. Quoting clearly any results you use, show that any nontrivial element of $\mathcal{M}$ of finite order has precisely 2 fixed points.

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• Paper 3, Section II, $7 \mathrm{D}$

  Groups | Part IA, 2021

  Let $G$ be a finite group of order $n$. Show that $G$ is isomorphic to a subgroup $H$ of $S_{n}$, the symmetric group of degree $n$. Furthermore show that this isomorphism can be chosen so that any nontrivial element of $H$ has no fixed points.

  Suppose $n$ is even. Prove that $G$ contains an element of order 2 .

  What does it mean for an element of $S_{m}$ to be odd? Suppose $H$ is a subgroup of $S_{m}$ for some $m$, and $H$ contains an odd element. Prove that precisely half of the elements of $H$ are odd.

  Now suppose $n=4 k+2$ for some positive integer $k$. Prove that $G$ is not simple. [Hint: Consider the sign of an element of order 2.]

  Can a nonabelian group of even order be simple?

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• Paper 3, Section II, D

  Groups | Part IA, 2021

  (a) Let $A$ be an abelian group (not necessarily finite). We define the generalised dihedral group to be the set of pairs

  $$D(A)=\{(a, \varepsilon): a \in A, \varepsilon=\pm 1\}$$

  with multiplication given by

  $$(a, \varepsilon)(b, \eta)=\left(a b^{\varepsilon}, \varepsilon \eta\right)$$

  The identity is $(e, 1)$ and the inverse of $(a, \varepsilon)$ is $\left(a^{-\varepsilon}, \varepsilon\right)$. You may assume that this multiplication defines a group operation on $D(A)$.

  (i) Identify $A$ with the set of all pairs in which $\varepsilon=+1$. Show that $A$ is a subgroup of $D(A)$. By considering the index of $A$ in $D(A)$, or otherwise, show that $A$ is a normal subgroup of $D(A)$.

  (ii) Show that every element of $D(A)$ not in $A$ has order 2 . Show that $D(A)$ is abelian if and only if $a^{2}=e$ for all $a \in A$. If $D(A)$ is non-abelian, what is the centre of $D(A) ?$ Justify your answer.

  (b) Let $\mathrm{O}(2)$ denote the group of $2 \times 2$ orthogonal matrices. Show that all elements of $\mathrm{O}(2)$ have determinant 1 or $-1$. Show that every element of $\mathrm{SO}(2)$ is a rotation. Let $J=\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right)$. Show that $\mathrm{O}(2)$ decomposes as a union $\mathrm{SO}(2) \cup \operatorname{SO}(2) J$.

  [You may assume standard properties of determinants.]

  (c) Let $B$ be the (abelian) group $\{z \in \mathbb{C}:|z|=1\}$, with multiplication of complex numbers as the group operation. Write down, without proof, isomorphisms $\mathrm{SO}(2) \cong B \cong \mathbb{R} / \mathbb{Z}$ where $\mathbb{R}$ denotes the additive group of real numbers and $\mathbb{Z}$ the subgroup of integers. Deduce that $\mathrm{O}(2) \cong D(B)$, the generalised dihedral group defined in part (a).

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• Paper 3, Section II, D

  Groups | Part IA, 2021

  (a) Let $G$ be a finite group acting on a set $X$. For $x \in X$, define the orbit $\operatorname{Orb}(x)$ and the stabiliser $\operatorname{Stab}(x)$ of $x$. Show that $\operatorname{Stab}(x)$ is a subgroup of $G$. State and prove the orbit-stabiliser theorem.

  (b) Let $n \geqslant k \geqslant 1$ be integers. Let $G=S_{n}$, the symmetric group of degree $n$, and $X$ be the set of all ordered $k$-tuples $\left(x_{1}, \ldots, x_{k}\right)$ with $x_{i} \in\{1,2, \ldots, n\}$. Then $G$ acts on $X$, where the action is defined by $\sigma\left(x_{1}, \ldots, x_{k}\right)=\left(\sigma\left(x_{1}\right), \ldots, \sigma\left(x_{k}\right)\right)$ for $\sigma \in S_{n}$ and $\left(x_{1}, \ldots, x_{k}\right) \in X$. For $x=(1,2, \ldots, k) \in X$, determine $\operatorname{Orb}(x)$ and $\operatorname{Stab}(x)$ and verify that the orbit-stabiliser theorem holds in this case.

  (c) We say that $G$ acts doubly transitively on $X$ if, whenever $\left(x_{1}, x_{2}\right)$ and $\left(y_{1}, y_{2}\right)$ are elements of $X \times X$ with $x_{1} \neq x_{2}$ and $y_{1} \neq y_{2}$, there exists some $g \in G$ such that $g x_{1}=y_{1}$ and $g x_{2}=y_{2}$.

  Assume that $G$ is a finite group that acts doubly transitively on $X$, and let $x \in X$. Show that if $H$ is a subgroup of $G$ that properly contains $\operatorname{Stab}(x)($ that is, $\operatorname{Stab}(x) \subseteq H$ but $\operatorname{Stab}(x) \neq H)$ then the action of $H$ on $X$ is transitive. Deduce that $H=G$.

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• Paper 3, Section II, D

  Groups | Part IA, 2021

  (a) Let $x$ be an element of a finite group $G$. Define the order of $x$ and the order of $G$. State and prove Lagrange's theorem. Deduce that the order of $x$ divides the order of $G$.

  (b) If $G$ is a group of order $n$, and $d$ is a divisor of $n$ where $d<n$, is it always true that $G$ must contain an element of order $d$ ? Justify your answer.

  (c) Denote the cyclic group of order $m$ by $C_{m}$.

  (i) Prove that if $m$ and $n$ are coprime then the direct product $C_{m} \times C_{n}$ is cyclic.

  (ii) Show that if a finite group $G$ has all non-identity elements of order 2 , then $G$ is isomorphic to $C_{2} \times \cdots \times C_{2}$. [The direct product theorem may be used without proof.]

  (d) Let $G$ be a finite group and $H$ a subgroup of $G$.

  (i) Let $x$ be an element of order $d$ in $G$. If $r$ is the least positive integer such that $x^{r} \in H$, show that $r$ divides $d$.

  (ii) Suppose further that $H$ has index $n$. If $x \in G$, show that $x^{k} \in H$ for some $k$ such that $0<k \leqslant n$. Is it always the case that the least positive such $k$ is a factor of $n$ ? Justify your answer.

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Vector Calculus

• Paper 3, Section I, B

  Vector Calculus | Part IA, 2021

  (a) What is meant by an antisymmetric tensor of second rank? Show that if a second rank tensor is antisymmetric in one Cartesian coordinate system, it is antisymmetric in every Cartesian coordinate system.

  (b) Consider the vector field $\mathbf{F}=(y, z, x)$ and the second rank tensor defined by $T_{i j}=\partial F_{i} / \partial x_{j}$. Calculate the components of the antisymmetric part of $T_{i j}$ and verify that it equals $-(1 / 2) \epsilon_{i j k} B_{k}$, where $\epsilon_{i j k}$ is the alternating tensor and $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{F}$.

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• Paper 3, Section I, B

  Vector Calculus | Part IA, 2021

  (a) Prove that

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

  where $\mathbf{A}$ and $\mathbf{B}$ are differentiable vector fields and $\psi$ is a differentiable scalar field.

  (b) Find the solution of $\nabla^{2} u=16 r^{2}$ on the two-dimensional domain $\mathcal{D}$ when

  (i) $\mathcal{D}$ is the unit disc $0 \leqslant r \leqslant 1$, and $u=1$ on $r=1$;

  (ii) $\mathcal{D}$ is the annulus $1 \leqslant r \leqslant 2$, and $u=1$ on both $r=1$ and $r=2$.

  [Hint: the Laplacian in plane polar coordinates is:

  $$\left.\nabla^{2} u=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}} . \quad\right]$$

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• Paper 3, Section II, B

  Vector Calculus | Part IA, 2021

  For a given charge distribution $\rho(\mathbf{x}, t)$ and current distribution $\mathbf{J}(\mathbf{x}, t)$ in $\mathbb{R}^{3}$, the electric and magnetic fields, $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$, satisfy Maxwell's equations, which in suitable units, read

  $$\begin{aligned} \boldsymbol{\nabla} \cdot \mathbf{E}=\rho, & \boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ \boldsymbol{\nabla} \cdot \mathbf{B}=0, & \boldsymbol{\nabla} \times \mathbf{B}=\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t} \end{aligned}$$

  The Poynting vector $\mathbf{P}$ is defined as $\mathbf{P}=\mathbf{E} \times \mathbf{B}$.

  (a) For a closed surface $\mathcal{S}$ around a volume $\mathcal{V}$, show that

  $$\int_{\mathcal{S}} \mathbf{P} \cdot d \mathbf{S}=-\int_{\mathcal{V}} \mathbf{E} \cdot \mathbf{J} d V-\frac{\partial}{\partial t} \int_{\mathcal{V}} \frac{|\mathbf{E}|^{2}+|\mathbf{B}|^{2}}{2} d V$$

  (b) Suppose $\mathbf{J}=\mathbf{0}$ and consider an electromagnetic wave

  $$\mathbf{E}=E_{0} \hat{\mathbf{y}} \cos (k x-\omega t) \quad \text { and } \quad \mathbf{B}=B_{0} \hat{\mathbf{z}} \cos (k x-\omega t)$$

  where $E_{0}, B_{0}, k$ and $\omega$ are positive constants. Show that these fields satisfy Maxwell's equations for appropriate $E_{0}, \omega$, and $\rho$.

  Confirm the wave satisfies the integral identity $(*)$ by considering its propagation through a box $\mathcal{V}$, defined by $0 \leqslant x \leqslant \pi /(2 k), 0 \leqslant y \leqslant L$, and $0 \leqslant z \leqslant L$.

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• Paper 3, Section II, B

  Vector Calculus | Part IA, 2021

  (a) By a suitable change of variables, calculate the volume enclosed by the ellipsoid $x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1$, where $a, b$, and $c$ are constants.

  (b) Suppose $T_{i j}$ is a second rank tensor. Use the divergence theorem to show that

  $$\int_{\mathcal{S}} T_{i j} n_{j} d S=\int_{\mathcal{V}} \frac{\partial T_{i j}}{\partial x_{j}} d V$$

  where $\mathcal{S}$ is a closed surface, with unit normal $n_{j}$, and $\mathcal{V}$ is the volume it encloses.

  [Hint: Consider $e_{i} T_{i j}$ for a constant vector $\left.e_{i} .\right]$

  (c) A half-ellipsoidal membrane $\mathcal{S}$ is described by the open surface $4 x^{2}+4 y^{2}+z^{2}=4$, with $z \geqslant 0$. At a given instant, air flows beneath the membrane with velocity $\mathbf{u}=$ $(-y, x, \alpha)$, where $\alpha$ is a constant. The flow exerts a force on the membrane given by

  $$F_{i}=\int_{\mathcal{S}} \beta^{2} u_{i} u_{j} n_{j} d S$$

  where $\beta$ is a constant parameter.

  Show the vector $a_{i}=\partial\left(u_{i} u_{j}\right) / \partial x_{j}$ can be rewritten as $\mathbf{a}=-(x, y, 0)$.

  Hence use $(*)$ to calculate the force $F_{i}$ on the membrane.

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• Paper 3, Section II, B

  Vector Calculus | Part IA, 2021

  (a) By considering an appropriate double integral, show that

  $$\int_{0}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{4 a}}$$

  where $a>0$.

  (b) Calculate $\int_{0}^{1} x^{y} d y$, treating $x$ as a constant, and hence show that

  $$\int_{0}^{\infty} \frac{\left(e^{-u}-e^{-2 u}\right)}{u} d u=\log 2$$

  (c) Consider the region $\mathcal{D}$ in the $x-y$ plane enclosed by $x^{2}+y^{2}=4, y=1$, and $y=\sqrt{3} x$ with $1<y<\sqrt{3} x$.

  Sketch $\mathcal{D}$, indicating any relevant polar angles.

  A surface $\mathcal{S}$ is given by $z=x y /\left(x^{2}+y^{2}\right)$. Calculate the volume below this surface and above $\mathcal{D}$.

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• Paper 3, Section II, B

  Vector Calculus | Part IA, 2021

  (a) Given a space curve $\mathbf{r}(t)=(x(t), y(t), z(t)$, with $t$ a parameter (not necessarily arc-length), give mathematical expressions for the unit tangent, unit normal, and unit binormal vectors.

  (b) Consider the closed curve given by

  $$x=2 \cos ^{3} t, \quad y=\sin ^{3} t, \quad z=\sqrt{3} \sin ^{3} t$$

  where $t \in[0,2 \pi)$.

  Show that the unit tangent vector $\mathbf{T}$ may be written as

  $$\mathbf{T}=\pm \frac{1}{2}(-2 \cos t, \sin t, \sqrt{3} \sin t)$$

  with each sign associated with a certain range of $t$, which you should specify.

  Calculate the unit normal and the unit binormal vectors, and hence deduce that the curve lies in a plane.

  (c) A closed space curve $\mathcal{C}$ lies in a plane with unit normal $\mathbf{n}=(a, b, c)$. Use Stokes' theorem to prove that the planar area enclosed by $\mathcal{C}$ is the absolute value of the line integral

  $$\frac{1}{2} \int_{\mathcal{C}}(b z-c y) d x+(c x-a z) d y+(a y-b x) d z$$

  Hence show that the planar area enclosed by the curve given by $(*)$ is $(3 / 2) \pi$.

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