• # Paper 3, Section I, D

Let $G$ be a group, and suppose the centre of $G$ is trivial. If $p$ divides $|G|$, show that $G$ has a non-trivial conjugacy class whose order is prime to $p$.

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• # Paper 3, Section I, D

Let $G=\mathbb{Q}$ be the rational numbers, with addition as the group operation. Let $x, y$ be non-zero elements of $G$, and let $N \leqslant G$ be the subgroup they generate. Show that $N$ is isomorphic to $\mathbb{Z}$.

Find non-zero elements $x, y \in \mathbb{R}$ which generate a subgroup that is not isomorphic to $\mathbb{Z}$.

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• # Paper 3, Section II, D

(a) Let $G$ be a group, and $N$ a subgroup of $G$. Define what it means for $N$ to be normal in $G$, and show that if $N$ is normal then $G / N$ naturally has the structure of a group.

(b) For each of (i)-(iii) below, give an example of a non-trivial finite group $G$ and non-trivial normal subgroup $N \leqslant G$ satisfying the stated properties.

(i) $G / N \times N \simeq G$.

(ii) There is no group homomorphism $G / N \rightarrow G$ such that the composite $G / N \rightarrow G \rightarrow G / N$ is the identity.

(iii) There is a group homomorphism $i: G / N \rightarrow G$ such that the composite $G / N \rightarrow G \rightarrow G / N$ is the identity, but the map

$G / N \times N \rightarrow G, \quad(g N, n) \mapsto i(g N) n$

is not a group homomorphism.

Show also that for any $N \leqslant G$ satisfying (iii), this map is always a bijection.

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• # Paper 3, Section II, D

Let $p$ be a prime number, and $G=G L_{2}\left(\mathbb{F}_{p}\right)$, the group of $2 \times 2$ invertible matrices with entries in the field $\mathbb{F}_{p}$ of integers modulo $p$.

The group $G$ acts on $X=\mathbb{F}_{p} \cup\{\infty\}$ by MÃ¶bius transformations,

$\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \cdot z=\frac{a z+b}{c z+d}$

(i) Show that given any distinct $x, y, z \in X$ there exists $g \in G$ such that $g \cdot 0=x$, $g \cdot 1=y$ and $g \cdot \infty=z$. How many such $g$ are there?

(ii) $G$ acts on $X \times X \times X$ by $g \cdot(x, y, z)=(g \cdot x, g \cdot y, g \cdot z)$. Describe the orbits, and for each orbit, determine its stabiliser, and the orders of the orbit and stabiliser.

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• # Paper 3, Section II, D

Let $p$ be a prime number. Let $G$ be a group such that every non-identity element of $G$ has order $p$.

(i) Show that if $|G|$ is finite, then $|G|=p^{n}$ for some $n$. [You must prove any theorems that you use.]

(ii) Show that if $H \leqslant G$, and $x \notin H$, then $\langle x\rangle \cap H=\{1\}$.

Hence show that if $G$ is abelian, and $|G|$ is finite, then $G \simeq C_{p} \times \cdots \times C_{p}$.

(iii) Let $G$ be the set of all $3 \times 3$ matrices of the form

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

where $a, b, x \in \mathbb{F}_{p}$ and $\mathbb{F}_{p}$ is the field of integers modulo $p$. Show that every nonidentity element of $G$ has order $p$ if and only if $p>2$. [You may assume that $G$ is a subgroup of the group of all $3 \times 3$ invertible matrices.]

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• # Paper 3, Section II, D

Let $S_{n}$ be the group of permutations of $\{1, \ldots, n\}$, and suppose $n$ is even, $n \geqslant 4$.

Let $g=(12) \in S_{n}$, and $h=(12)(34) \ldots(n-1 n) \in S_{n}$.

(i) Compute the centraliser of $g$, and the orders of the centraliser of $g$ and of the centraliser of $h$.

(ii) Now let $n=6$. Let $G$ be the group of all symmetries of the cube, and $X$ the set of faces of the cube. Show that the action of $G$ on $X$ makes $G$ isomorphic to the centraliser of $h$ in $S_{6}$. [Hint: Show that $-1 \in G$ permutes the faces of the cube according to $h$.]

Show that $G$ is also isomorphic to the centraliser of $g$ in $S_{6}$.

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• # Paper 3, Section I, A

Let $\mathbf{F}(\mathbf{x})$ be a vector field defined everywhere on the domain $G \subset \mathbb{R}^{3}$.

(a) Suppose that $\mathbf{F}(\mathbf{x})$ has a potential $\phi(\mathbf{x})$ such that $\mathbf{F}(\mathbf{x})=\nabla \phi(\mathbf{x})$ for $\mathbf{x} \in G$. Show that

$\int_{\gamma} \mathbf{F} \cdot \mathbf{d} \mathbf{x}=\phi(\mathbf{b})-\phi(\mathbf{a})$

for any smooth path $\gamma$ from a to $\mathbf{b}$ in $G$. Show further that necessarily $\nabla \times \mathbf{F}=\mathbf{0}$ on $G$.

(b) State a condition for $G$ which ensures that $\nabla \times \mathbf{F}=\mathbf{0}$ implies $\int_{\gamma} \mathbf{F} \cdot \mathbf{d x}$ is pathindependent.

(c) Compute the line integral $\oint_{\gamma} \mathbf{F} \cdot \mathbf{d} \mathbf{x}$ for the vector field

$\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} \frac{-y}{x^{2}+y^{2}} \\ \frac{x}{x^{2}+y^{2}} \\ 0 \end{array}\right)$

where $\gamma$ denotes the anti-clockwise path around the unit circle in the $(x, y)$-plane. Compute $\nabla \times \mathbf{F}$ and comment on your result in the light of (b).

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• # Paper 3, Section I, A

(a) For $\mathbf{x} \in \mathbb{R}^{n}$ and $r=|\mathbf{x}|$, show that

$\frac{\partial r}{\partial x_{i}}=\frac{x_{i}}{r}$

(b) Use index notation and your result in (a), or otherwise, to compute

(i) $\nabla \cdot(f(r) \mathbf{x})$, and

(ii) $\nabla \times(f(r) \mathbf{x})$ for $n=3$.

(c) Show that for each $n \in \mathbb{N}$ there is, up to an arbitrary constant, just one vector field $\mathbf{F}(\mathbf{x})$ of the form $f(r) \mathbf{x}$ such that $\nabla \cdot \mathbf{F}(\mathbf{x})=0$ everywhere on $\mathbb{R}^{n} \backslash\{\mathbf{0}\}$, and determine $\mathbf{F}$.

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• # Paper 3, Section II, 11A

(i) Starting with Poisson's equation in $\mathbb{R}^{3}$,

$\nabla^{2} \phi(\mathbf{x})=f(\mathbf{x})$

derive Gauss' flux theorem

$\int_{V} f(\mathbf{x}) d V=\int_{\partial V} \mathbf{F}(\mathbf{x}) \cdot \mathbf{d} \mathbf{S}$

for $\mathbf{F}(\mathbf{x})=\nabla \phi(\mathbf{x})$ and for any volume $V \subseteq \mathbb{R}^{3}$.

(ii) Let

$I=\int_{S} \frac{\mathbf{x} \cdot \mathbf{d} \mathbf{S}}{|\mathbf{x}|^{3}} .$

Show that $I=4 \pi$ if $S$ is the sphere $|\mathbf{x}|=R$, and that $I=0$ if $S$ bounds a volume that does not contain the origin.

(iii) Show that the electric field defined by

$\mathbf{E}(\mathbf{x})=\frac{q}{4 \pi \epsilon_{0}} \frac{\mathbf{x}-\mathbf{a}}{|\mathbf{x}-\mathbf{a}|^{3}}, \quad \mathbf{x} \neq \mathbf{a}$

satisfies

$\int_{\partial V} \mathbf{E} \cdot \mathbf{d} \mathbf{S}= \begin{cases}0 & \text { if } \mathbf{a} \notin V \\ \frac{q}{\epsilon_{0}} & \text { if } \mathbf{a} \in V\end{cases}$

where $\partial V$ is a surface bounding a closed volume $V$ and $\mathbf{a} \notin \partial V$, and where the electric charge $q$ and permittivity of free space $\epsilon_{0}$ are constants. This is Gauss' law for a point electric charge.

(iv) Assume that $f(\mathbf{x})$ is spherically symmetric around the origin, i.e., it is a function only of $|\mathbf{x}|$. Assume that $\mathbf{F}(\mathbf{x})$ is also spherically symmetric. Show that $\mathbf{F}(\mathbf{x})$ depends only on the values of $f$ inside the sphere with radius $|\mathbf{x}|$ but not on the values of $f$ outside this sphere.

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• # Paper 3, Section II, A

(a) Show that any rank 2 tensor $t_{i j}$ can be written uniquely as a sum of two rank 2 tensors $s_{i j}$ and $a_{i j}$ where $s_{i j}$ is symmetric and $a_{i j}$ is antisymmetric.

(b) Assume that the rank 2 tensor $t_{i j}$ is invariant under any rotation about the $z$-axis, as well as under a rotation of angle $\pi$ about any axis in the $(x, y)$-plane through the origin.

(i) Show that there exist $\alpha, \beta \in \mathbb{R}$ such that $t_{i j}$ can be written as

$t_{i j}=\alpha \delta_{i j}+\beta \delta_{i 3} \delta_{j 3} .$

(ii) Is there some proper subgroup of the rotations specified above for which the result $(*)$ still holds if the invariance of $t_{i j}$ is restricted to this subgroup? If so, specify the smallest such subgroup.

(c) The array of numbers $d_{i j k}$ is such that $d_{i j k} s_{i j}$ is a vector for any symmetric matrix $s_{i j}$.

(i) By writing $d_{i j k}$ as a sum of $d_{i j k}^{s}$ and $d_{i j k}^{a}$ with $d_{i j k}^{s}=d_{j i k}^{s}$ and $d_{i j k}^{a}=-d_{j i k}^{a}$, show that $d_{i j k}^{s}$ is a rank 3 tensor. [You may assume without proof the Quotient Theorem for tensors.]

(ii) Does $d_{i j k}^{a}$ necessarily have to be a tensor? Justify your answer.

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• # Paper 3, Section II, A

(a) State Stokes' Theorem for a surface $S$ with boundary $\partial S$.

(b) Let $S$ be the surface in $\mathbb{R}^{3}$ given by $z^{2}=1+x^{2}+y^{2}$ where $\sqrt{2} \leqslant z \leqslant \sqrt{5}$. Sketch the surface $S$ and find the surface element $\mathbf{d} \mathbf{S}$ with respect to the Cartesian coordinates $x$ and $y$.

(c) Compute $\nabla \times \mathbf{F}$ for the vector field

$\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} -y \\ x \\ x y(x+y) \end{array}\right)$

and verify Stokes' Theorem for $\mathbf{F}$ on the surface $S$.

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• # Paper 3, Section II, A

The surface $C$ in $\mathbb{R}^{3}$ is given by $z^{2}=x^{2}+y^{2}$.

(a) Show that the vector field

$\mathbf{F}(\mathbf{x})=\left(\begin{array}{l} x \\ y \\ z \end{array}\right)$

is tangent to the surface $C$ everywhere.

(b) Show that the surface integral $\int_{S} \mathbf{F} \cdot \mathbf{d} \mathbf{S}$ is a constant independent of $S$ for any surface $S$ which is a subset of $C$, and determine this constant.

(c) The volume $V$ in $\mathbb{R}^{3}$ is bounded by the surface $C$ and by the cylinder $x^{2}+y^{2}=1$. Sketch $V$ and compute the volume integral

$\int_{V} \nabla \cdot \mathbf{F} d V$

directly by integrating over $V$.

(d) Use the Divergence Theorem to verify the result you obtained in part (b) for the integral $\int_{S} \mathbf{F} \cdot \mathbf{d} \mathbf{S}$, where $S$ is the portion of $C$ lying in $-1 \leqslant z \leqslant 1$.

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