• # Paper 3 , Section II, E

State Lagrange's theorem. Show that the order of an element $x$ in a finite group $G$ is finite and divides the order of $G$.

State Cauchy's theorem.

List all groups of order 8 up to isomorphism. Carefully justify that the groups on your list are pairwise non-isomorphic and that any group of order 8 is isomorphic to one on your list. [You may use without proof the Direct Product Theorem and the description of standard groups in terms of generators satisfying certain relations.]

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• # Paper 3, Section I, E

What does it mean to say that $H$ is a normal subgroup of the group $G$ ? For a normal subgroup $H$ of $G$ define the quotient group $G / H$. [You do not need to verify that $G / H$ is a group.]

State the Isomorphism Theorem.

Let

$G=\left\{\left(\begin{array}{ll} a & b \\ 0 & d \end{array}\right) \mid a, b, d \in \mathbb{R}, a d \neq 0\right\}$

be the group of $2 \times 2$ invertible upper-triangular real matrices. By considering a suitable homomorphism, show that the subset

$H=\left\{\left(\begin{array}{ll} 1 & b \\ 0 & 1 \end{array}\right) \mid b \in \mathbb{R}\right\}$

of $G$ is a normal subgroup of $G$ and identify the quotient $G / H$.

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• # Paper 3, Section I, E

Let $w_{1}, w_{2}, w_{3}$ be distinct elements of $\mathbb{C} \cup\{\infty\}$. Write down the Möbius map $f$ that sends $w_{1}, w_{2}, w_{3}$ to $\infty, 0,1$, respectively. [Hint: You need to consider four cases.]

Now let $w_{4}$ be another element of $\mathbb{C} \cup\{\infty\}$ distinct from $w_{1}, w_{2}, w_{3}$. Define the cross-ratio $\left[w_{1}, w_{2}, w_{3}, w_{4}\right]$ in terms of $f$.

Prove that there is a circle or line through $w_{1}, w_{2}, w_{3}$ and $w_{4}$ if and only if the cross-ratio $\left[w_{1}, w_{2}, w_{3}, w_{4}\right]$ is real.

[You may assume without proof that Möbius maps map circles and lines to circles and lines and also that there is a unique circle or line through any three distinct points of $\mathbb{C} \cup\{\infty\} .]$

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

(a) Let $G$ be a finite group acting on a finite set $X$. State the Orbit-Stabiliser theorem. [Define the terms used.] Prove that

$\sum_{x \in X}|\operatorname{Stab}(x)|=n|G|$

where $n$ is the number of distinct orbits of $X$ under the action of $G$.

Let $S=\{(g, x) \in G \times X: g \cdot x=x\}$, and for $g \in G$, let $\operatorname{Fix}(g)=\{x \in X: g \cdot x=x\}$.

Show that

$|S|=\sum_{x \in X}|\operatorname{Stab}(x)|=\sum_{g \in G}|\operatorname{Fix}(g)|$

and deduce that

$n=\frac{1}{|G|} \sum_{g \in G}|\operatorname{Fix}(g)|$

(b) Let $H$ be the group of rotational symmetries of the cube. Show that $H$ has 24 elements. [If your proof involves calculating stabilisers, then you must carefully verify such calculations.]

Using $(*)$, find the number of distinct ways of colouring the faces of the cube red, green and blue, where two colourings are distinct if one cannot be obtained from the other by a rotation of the cube. [A colouring need not use all three colours.]

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• # Paper 3, Section II, E

Prove that every element of the symmetric group $S_{n}$ is a product of transpositions. [You may assume without proof that every permutation is the product of disjoint cycles.]

(a) Define the sign of a permutation in $S_{n}$, and prove that it is well defined. Define the alternating group $A_{n}$.

(b) Show that $S_{n}$ is generated by the set $\left\{\left(\begin{array}{lll}1 & 2\end{array}\right),\left(\begin{array}{llll}1 & 2 & 3 & \ldots\end{array}\right)\right\}$.

Given $1 \leqslant k, prove that the set $\left\{\left(\begin{array}{ll}1 & \left.1+k),\left(\begin{array}{lll}1 & 2 & 3\end{array} \ldots n\right)\right\} \text { generates } S_{n} \text { if and only }\end{array}\right.\right.$ if $k$ and $n$ are coprime.

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• # Paper 3, Section II, E

Let $N$ be a normal subgroup of a finite group $G$ of prime index $p=|G: N|$.

By considering a suitable homomorphism, show that if $H$ is a subgroup of $G$ that is not contained in $N$, then $H \cap N$ is a normal subgroup of $H$ of index $p$.

Let $C$ be a conjugacy class of $G$ that is contained in $N$. Prove that $C$ is either a conjugacy class in $N$ or is the disjoint union of $p$ conjugacy classes in $N$.

[You may use standard theorems without proof.]

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• # Paper 3, Section $I$, B

(a) The two sets of basis vectors $\mathbf{e}_{i}$ and $\mathbf{e}_{i}^{\prime}$ (where $i=1,2,3$ ) are related by

$\mathbf{e}_{i}^{\prime}=R_{i j} \mathbf{e}_{j}$

where $R_{i j}$ are the entries of a rotation matrix. The components of a vector $\mathbf{v}$ with respect to the two bases are given by

$\mathbf{v}=v_{i} \mathbf{e}_{i}=v_{i}^{\prime} \mathbf{e}_{i}^{\prime}$

Derive the relationship between $v_{i}$ and $v_{i}^{\prime}$.

(b) Let $\mathbf{T}$ be a $3 \times 3$ array defined in each (right-handed orthonormal) basis. Using part (a), state and prove the quotient theorem as applied to $\mathbf{T}$.

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• # Paper 3, Section I, B

Use the change of variables $x=r \cosh \theta, y=r \sinh \theta$ to evaluate

$\int_{A} y d x d y$

where $A$ is the region of the $x y$-plane bounded by the two line segments:

\begin{aligned} &y=0, \quad 0 \leqslant x \leqslant 1 \\ &5 y=3 x, \quad 0 \leqslant x \leqslant \frac{5}{4} \end{aligned}

and the curve

$x^{2}-y^{2}=1, \quad x \geqslant 1$

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• # Paper 3, Section II, B

Let $S$ be a piecewise smooth closed surface in $\mathbb{R}^{3}$ which is the boundary of a volume $V$.

(a) The smooth functions $\phi$ and $\phi_{1}$ defined on $\mathbb{R}^{3}$ satisfy

$\nabla^{2} \phi=\nabla^{2} \phi_{1}=0$

in $V$ and $\phi(\mathbf{x})=\phi_{1}(\mathbf{x})=f(\mathbf{x})$ on $S$. By considering an integral of $\boldsymbol{\nabla} \psi \cdot \boldsymbol{\nabla} \psi$, where $\psi=\phi-\phi_{1}$, show that $\phi_{1}=\phi$.

(b) The smooth function $u$ defined on $\mathbb{R}^{3}$ satisfies $u(\mathbf{x})=f(\mathbf{x})+C$ on $S$, where $f$ is the function in part (a) and $C$ is constant. Show that

$\int_{V} \nabla u \cdot \nabla u d V \geqslant \int_{V} \nabla \phi \cdot \nabla \phi d V$

where $\phi$ is the function in part (a). When does equality hold?

(c) The smooth function $w(\mathbf{x}, t)$ satisfies

$\nabla^{2} w=\frac{\partial w}{\partial t}$

in $V$ and $\frac{\partial w}{\partial t}=0$ on $S$ for all $t$. Show that

$\frac{d}{d t} \int_{V} \nabla w \cdot \nabla w d V \leqslant 0$

with equality only if $\nabla^{2} w=0$ in $V$.

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• # Paper 3, Section II, B

(a) Let $\mathbf{x}=\mathbf{r}(s)$ be a smooth curve parametrised by arc length $s$. Explain the meaning of the terms in the equation

$\frac{d \mathbf{t}}{d s}=\kappa \mathbf{n},$

where $\kappa(s)$ is the curvature of the curve.

Now let $\mathbf{b}=\mathbf{t} \times \mathbf{n}$. Show that there is a scalar $\tau(s)$ (the torsion) such that

$\frac{d \mathbf{b}}{d s}=-\tau \mathbf{n}$

and derive an expression involving $\kappa$ and $\tau$ for $\frac{d \mathbf{n}}{d s}$.

(b) Given a (nowhere zero) vector field $\mathbf{F}$, the field lines, or integral curves, of $\mathbf{F}$ are the curves parallel to $\mathbf{F}(\mathbf{x})$ at each point $\mathbf{x}$. Show that the curvature $\kappa$ of the field lines of $\mathbf{F}$ satisfies

$\frac{\mathbf{F} \times(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{F}}{F^{3}}=\pm \kappa \mathbf{b}$

where $F=|\mathbf{F}|$.

(c) Use $(*)$ to find an expression for the curvature at the point $(x, y, z)$ of the field lines of $\mathbf{F}(x, y, z)=(x, y,-z)$.

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• # Paper 3, Section II, B

By a suitable choice of $\mathbf{u}$ in the divergence theorem

$\int_{V} \nabla \cdot \mathbf{u} d V=\int_{S} \mathbf{u} \cdot d \mathbf{S}$

show that

$\int_{V} \nabla \phi d V=\int_{S} \phi d \mathbf{S}$

for any continuously differentiable function $\phi$.

For the curved surface of the cone

$\mathbf{x}=(r \cos \theta, r \sin \theta, \sqrt{3} r), \quad 0 \leqslant \sqrt{3} r \leqslant 1, \quad 0 \leqslant \theta \leqslant 2 \pi$

show that $d \mathbf{S}=(\sqrt{3} \cos \theta, \sqrt{3} \sin \theta,-1) r d r d \theta$.

Verify that $(*)$ holds for this cone and $\phi(x, y, z)=z^{2}$.

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• # Paper 3, Section II, B

(a) The time-dependent vector field $\mathbf{F}$ is related to the vector field $\mathbf{B}$ by

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

where $\mathbf{z}=t \mathbf{x}$. Show that

$(\mathbf{x} \cdot \nabla) \mathbf{F}=t \frac{\partial \mathbf{F}}{\partial t} \text {. }$

(b) The vector fields $\mathbf{B}$ and $\mathbf{A}$ satisfy $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. Show that $\boldsymbol{\nabla} \cdot \mathbf{B}=0$.

(c) The vector field $\mathbf{B}$ satisfies $\boldsymbol{\nabla} \cdot \mathbf{B}=0$. Show that

$\mathbf{B}(\mathbf{x})=\nabla \times(\mathbf{D}(\mathbf{x}) \times \mathbf{x})$

where

$\mathbf{D}(\mathbf{x})=\int_{0}^{1} t \mathbf{B}(t \mathbf{x}) d t$

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