• # Paper 3, Section I, D

State and prove Lagrange's theorem.

Let $p$ be an odd prime number, and let $G$ be a finite group of order $2 p$ which has a normal subgroup of order 2 . Show that $G$ is a cyclic group.

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• # Paper 3, Section I, D

Let $G$ be a group, and let $H$ be a subgroup of $G$. Show that the following are equivalent.

(i) $a^{-1} b^{-1} a b \in H$ for all $a, b \in G$.

(ii) $H$ is a normal subgroup of $G$ and $G / H$ is abelian.

Hence find all abelian quotient groups of the dihedral group $D_{10}$ of order 10 .

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• # Paper 3, Section II, $7 \mathrm{D}$

State and prove the orbit-stabiliser theorem.

Let $p$ be a prime number, and $G$ be a finite group of order $p^{n}$ with $n \geqslant 1$. If $N$ is a non-trivial normal subgroup of $G$, show that $N \cap Z(G)$ contains a non-trivial element.

If $H$ is a proper subgroup of $G$, show that there is a $g \in G \backslash H$ such that $g^{-1} H g=H$.

[You may use Lagrange's theorem, provided you state it clearly.]

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• # Paper 3, Section II, D

Define the Möbius group $\mathcal{M}$ and its action on the Riemann sphere $\mathbb{C}_{\infty}$. [You are not required to verify the group axioms.] Show that there is a surjective group homomorphism $\phi: S L_{2}(\mathbb{C}) \rightarrow \mathcal{M}$, and find the kernel of $\phi .$

Show that if a non-trivial element of $\mathcal{M}$ has finite order, then it fixes precisely two points in $\mathbb{C}_{\infty}$. Hence show that any finite abelian subgroup of $\mathcal{M}$ is either cyclic or isomorphic to $C_{2} \times C_{2}$.

[You may use standard properties of the Möbius group, provided that you state them clearly.]

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• # Paper 3, Section II, D

Define the sign, $\operatorname{sgn}(\sigma)$, of a permutation $\sigma \in S_{n}$ and prove that it is well defined. Show that the function $\operatorname{sgn}: S_{n} \rightarrow\{1,-1\}$ is a homomorphism.

Show that there is an injective homomorphism $\psi: G L_{2}(\mathbb{Z} / 2 \mathbb{Z}) \rightarrow S_{4}$ such that $\operatorname{sgn} \circ \psi$ is non-trivial.

Show that there is an injective homomorphism $\phi: S_{n} \rightarrow G L_{n}(\mathbb{R})$ such that $\operatorname{det}(\phi(\sigma))=\operatorname{sgn}(\sigma) .$

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• # Paper 3, Section II, D

For each of the following, either give an example or show that none exists.

(i) A non-abelian group in which every non-trivial element has order $2 .$

(ii) A non-abelian group in which every non-trivial element has order 3 .

(iii) An element of $S_{9}$ of order 18 .

(iv) An element of $S_{9}$ of order 20 .

(v) A finite group which is not isomorphic to a subgroup of an alternating group.

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• # Paper 3, Section I, C

If $\mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right)$ and $\mathbf{w}=\left(w_{1}, w_{2}, w_{3}\right)$ are vectors in $\mathbb{R}^{3}$, show that $T_{i j}=v_{i} w_{j}$ defines a rank 2 tensor. For which choices of the vectors $\mathbf{v}$ and $\mathbf{w}$ is $T_{i j}$ isotropic?

Write down the most general isotropic tensor of rank 2 .

Prove that $\epsilon_{i j k}$ defines an isotropic rank 3 tensor.

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• # Paper 3, Section I, C

State the chain rule for the derivative of a composition $t \mapsto f(\mathbf{X}(t))$, where $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and $\mathbf{X}: \mathbb{R} \rightarrow \mathbb{R}^{n}$ are smooth $.$

Consider parametrized curves given by

$\mathbf{x}(t)=(x(t), y(t))=(a \cos t, a \sin t) .$

Calculate the tangent vector $\frac{d \mathbf{x}}{d t}$ in terms of $x(t)$ and $y(t)$. Given that $u(x, y)$ is a smooth function in the upper half-plane $\left\{(x, y) \in \mathbb{R}^{2} \mid y>0\right\}$ satisfying

$x \frac{\partial u}{\partial y}-y \frac{\partial u}{\partial x}=u$

deduce that

$\frac{d}{d t} u(x(t), y(t))=u(x(t), y(t))$

If $u(1,1)=10$, find $u(-1,1)$.

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• # Paper 3, Section II, C

(a) Let

$\mathbf{F}=(z, x, y)$

and let $C$ be a circle of radius $R$ lying in a plane with unit normal vector $(a, b, c)$. Calculate $\nabla \times \mathbf{F}$ and use this to compute $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$. Explain any orientation conventions which you use.

(b) Let $\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be a smooth vector field such that the matrix with entries $\frac{\partial F_{j}}{\partial x_{i}}$ is symmetric. Prove that $\oint_{C} \mathbf{F} \cdot d \mathbf{x}=0$ for every circle $C \subset \mathbb{R}^{3}$.

(c) Let $\mathbf{F}=\frac{1}{r}(x, y, z)$, where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ and let $C$ be the circle which is the intersection of the sphere $(x-5)^{2}+(y-3)^{2}+(z-2)^{2}=1$ with the plane $3 x-5 y-z=2$. Calculate $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$.

(d) Let $\mathbf{F}$ be the vector field defined, for $x^{2}+y^{2}>0$, by

$\mathbf{F}=\left(\frac{-y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}, z\right)$

Show that $\nabla \times \mathbf{F}=\mathbf{0}$. Let $C$ be the curve which is the intersection of the cylinder $x^{2}+y^{2}=1$ with the plane $z=x+200$. Calculate $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$.

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• # Paper 3, Section II, C

(a) For smooth scalar fields $u$ and $v$, derive the identity

$\nabla \cdot(u \nabla v-v \nabla u)=u \nabla^{2} v-v \nabla^{2} u$

and deduce that

\begin{aligned} \int_{\rho \leqslant|\mathbf{x}| \leqslant r}\left(v \nabla^{2} u-u \nabla^{2} v\right) d V=\int_{|\mathbf{x}|=r}\left(v \frac{\partial u}{\partial n}-u \frac{\partial v}{\partial n}\right) d S \\ &-\int_{|\mathbf{x}|=\rho}\left(v \frac{\partial u}{\partial n}-u \frac{\partial v}{\partial n}\right) d S \end{aligned}

Here $\nabla^{2}$ is the Laplacian, $\frac{\partial}{\partial n}=\mathbf{n} \cdot \nabla$ where $\mathbf{n}$ is the unit outward normal, and $d S$ is the scalar area element.

(b) Give the expression for $(\nabla \times \mathbf{V})_{i}$ in terms of $\epsilon_{i j k}$. Hence show that

$\nabla \times(\nabla \times \mathbf{V})=\nabla(\nabla \cdot \mathbf{V})-\nabla^{2} \mathbf{V}$

(c) Assume that if $\nabla^{2} \varphi=-\rho$, where $\varphi(\mathbf{x})=O\left(|\mathbf{x}|^{-1}\right)$ and $\nabla \varphi(\mathbf{x})=O\left(|\mathbf{x}|^{-2}\right)$ as $|\mathbf{x}| \rightarrow \infty$, then

$\varphi(\mathbf{x})=\int_{\mathbb{R}^{3}} \frac{\rho(\mathbf{y})}{4 \pi|\mathbf{x}-\mathbf{y}|} d V .$

The vector fields $\mathbf{B}$ and $\mathbf{J}$ satisfy

$\nabla \times \mathbf{B}=\mathbf{J}$

Show that $\nabla \cdot \mathbf{J}=0$. In the case that $\mathbf{B}=\nabla \times \mathbf{A}$, with $\nabla \cdot \mathbf{A}=0$, show that

$\mathbf{A}(\mathbf{x})=\int_{\mathbb{R}^{3}} \frac{\mathbf{J}(\mathbf{y})}{4 \pi|\mathbf{x}-\mathbf{y}|} d V$

and hence that

$\mathbf{B}(\mathbf{x})=\int_{\mathbb{R}^{3}} \frac{\mathbf{J}(\mathbf{y}) \times(\mathbf{x}-\mathbf{y})}{4 \pi|\mathbf{x}-\mathbf{y}|^{3}} d V$

Verify that $\mathbf{A}$ given by $(*)$ does indeed satisfy $\nabla \cdot \mathbf{A}=0$. [It may be useful to make a change of variables in the right hand side of $(*)$.]

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• # Paper 3, Section II, C

Define the Jacobian $J[\mathbf{u}]$ of a smooth mapping $\mathbf{u}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$. Show that if $\mathbf{V}$ is the vector field with components

$V_{i}=\frac{1}{3 !} \epsilon_{i j k} \epsilon_{a b c} \frac{\partial u_{a}}{\partial x_{j}} \frac{\partial u_{b}}{\partial x_{k}} u_{c}$

then $J[\mathbf{u}]=\nabla \cdot \mathbf{V}$. If $\mathbf{v}$ is another such mapping, state the chain rule formula for the derivative of the composition $\mathbf{w}(\mathbf{x})=\mathbf{u}(\mathbf{v}(\mathbf{x}))$, and hence give $J[\mathbf{w}]$ in terms of $J[\mathbf{u}]$ and $J[\mathbf{v}]$.

Let $\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be a smooth vector field. Let there be given, for each $t \in \mathbb{R}$, a smooth mapping $\mathbf{u}_{t}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ such that $\mathbf{u}_{t}(\mathbf{x})=\mathbf{x}+t \mathbf{F}(\mathbf{x})+o(t)$ as $t \rightarrow 0$. Show that

$J\left[\mathbf{u}_{t}\right]=1+t Q(x)+o(t)$

for some $Q(x)$, and express $Q$ in terms of $\mathbf{F}$. Assuming now that $\mathbf{u}_{t+s}(\mathbf{x})=\mathbf{u}_{t}\left(\mathbf{u}_{s}(\mathbf{x})\right)$, deduce that if $\nabla \cdot \mathbf{F}=0$ then $J\left[\mathbf{u}_{t}\right]=1$ for all $t \in \mathbb{R}$. What geometric property of the mapping $\mathbf{x} \mapsto \mathbf{u}_{t}(\mathbf{x})$ does this correspond to?

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• # Paper 3, Section II, C

What is a conservative vector field on $\mathbb{R}^{n}$ ?

State Green's theorem in the plane $\mathbb{R}^{2}$.

(a) Consider a smooth vector field $\mathbf{V}=(P(x, y), Q(x, y))$ defined on all of $\mathbb{R}^{2}$ which satisfies

$\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=0$

By considering

$F(x, y)=\int_{0}^{x} P\left(x^{\prime}, 0\right) d x^{\prime}+\int_{0}^{y} Q\left(x, y^{\prime}\right) d y^{\prime}$

or otherwise, show that $\mathbf{V}$ is conservative.

(b) Now let $\mathbf{V}=(1+\cos (2 \pi x+2 \pi y), 2+\cos (2 \pi x+2 \pi y))$. Show that there exists a smooth function $F(x, y)$ such that $\mathbf{V}=\nabla F$.

Calculate $\int_{C} \mathbf{V} \cdot d \mathbf{x}$, where $C$ is a smooth curve running from $(0,0)$ to $(m, n) \in \mathbb{Z}^{2}$. Deduce that there does not exist a smooth function $F(x, y)$ which satisfies $\mathbf{V}=\nabla F$ and which is, in addition, periodic with period 1 in each coordinate direction, i.e. $F(x, y)=F(x+1, y)=F(x, y+1)$.

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