Part IA, 2008, Paper 3

# Part IA, 2008, Paper 3

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3.I.1E

commentDefine the signature $\epsilon(\sigma)$ of a permutation $\sigma \in S_{n}$, and show that the map $\epsilon: S_{n} \rightarrow\{-1,1\}$ is a homomorphism.

Define the alternating group $A_{n}$, and prove that it is a subgroup of $S_{n}$. Is $A_{n}$ a normal subgroup of $S_{n}$ ? Justify your answer.

3.I.2E

commentWhat is the orthogonal group $O(n)$ ? What is the special orthogonal group $S O(n) ?$

Show that every element of the special orthogonal group $S O(3)$ has an eigenvector with eigenvalue 1 . Is this also true for every element of the orthogonal group $O(3)$ ? Justify your answer.

3.II $. 5 \mathrm{E} \quad$

commentFor a normal subgroup $H$ of a group $G$, explain carefully how to make the set of (left) cosets of $H$ into a group.

For a subgroup $H$ of a group $G$, show that the following are equivalent:

(i) $H$ is a normal subgroup of $G$;

(ii) there exist a group $K$ and a homomorphism $\theta: G \rightarrow K$ such that $H$ is the kernel of $\theta$.

Let $G$ be a finite group that has a proper subgroup $H$ of index $n$ (in other words, $|H|=|G| / n)$. Show that if $|G|>n$ ! then $G$ cannot be simple. [Hint: Let $G$ act on the set of left cosets of $H$ by left multiplication.]

3.II $. 7 \mathrm{E} \quad$

commentShow that every Möbius map may be expressed as a composition of maps of the form $z \mapsto z+a, z \mapsto \lambda z$ and $z \mapsto 1 / z$ (where $a$ and $\lambda$ are complex numbers).

Which of the following statements are true and which are false? Justify your answers.

(i) Every Möbius map that fixes $\infty$ may be expressed as a composition of maps of the form $z \mapsto z+a$ and $z \mapsto \lambda z$ (where $a$ and $\lambda$ are complex numbers).

(ii) Every Möbius map that fixes 0 may be expressed as a composition of maps of the form $z \mapsto \lambda z$ and $z \mapsto 1 / z$ (where $\lambda$ is a complex number).

(iii) Every Möbius map may be expressed as a composition of maps of the form $z \mapsto z+a$ and $z \mapsto 1 / z$ (where $a$ is a complex number).

3.II $. 8 \mathrm{E} \quad$

commentState and prove the orbit-stabilizer theorem. Deduce that if $x$ is an element of a finite group $G$ then the order of $x$ divides the order of $G$

Prove Cauchy's theorem, that if $p$ is a prime dividing the order of a finite group $G$ then $G$ contains an element of order $p$.

For which positive integers $n$ does there exist a group of order $n$ in which every element (apart from the identity) has order 2?

Give an example of an infinite group in which every element (apart from the identity) has order $2 .$

3.I.3C

commentA curve is given in terms of a parameter $t$ by

$\mathbf{x}(t)=\left(t-\frac{1}{3} t^{3}, t^{2}, t+\frac{1}{3} t^{3}\right)$

(i) Find the arc length of the curve between the points with $t=0$ and $t=1$.

(ii) Find the unit tangent vector at the point with parameter $t$, and show that the principal normal is orthogonal to the $z$ direction at each point on the curve.

3.I.4C

commentWhat does it mean to say that $T_{i j}$ transforms as a second rank tensor?

If $T_{i j}$ transforms as a second rank tensor, show that $\frac{\partial T_{i j}}{\partial x_{j}}$ transforms as a vector.

3.II.10C

commentFind the effect of a rotation by $\pi / 2$ about the $z$-axis on the tensor

$\left(\begin{array}{lll} S_{11} & S_{12} & S_{13} \\ S_{21} & S_{22} & S_{23} \\ S_{31} & S_{32} & S_{33} \end{array}\right)$

Hence show that the most general isotropic tensor of rank 2 is $\lambda \delta_{i j}$, where $\lambda$ is an arbitrary scalar.

Prove that there is no non-zero isotropic vector, and write down without proof the most general isotropic tensor of rank 3 .

Deduce that if $T_{i j k l}$ is an isotropic tensor then the following results hold, for some scalars $\mu$ and $\nu$ : (i) $\epsilon_{i j k} T_{i j k l}=0$; (ii) $\delta_{i j} T_{i j k l}=\mu \delta_{k l}$; (iii) $\epsilon_{i j m} T_{i j k l}=\nu \epsilon_{k l m}$.

Verify these three results in the case $T_{i j k l}=\alpha \delta_{i j} \delta_{k l}+\beta \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k}$, expressing $\mu$ and $\nu$ in terms of $\alpha, \beta$ and $\gamma$.

3.II.11C

commentLet $V$ be a volume in $\mathbb{R}^{3}$ bounded by a closed surface $S$.

(a) Let $f$ and $g$ be twice differentiable scalar fields such that $f=1$ on $S$ and $\nabla^{2} g=0$ in $V$. Show that

$\int_{V} \nabla f \cdot \nabla g d V=0$

(b) Let $V$ be the sphere $|\mathbf{x}| \leqslant a$. Evaluate the integral

$\int_{V} \nabla u \cdot \nabla v d V$

in the cases where $u$ and $v$ are given in spherical polar coordinates by: (i) $u=r, \quad v=r \cos \theta$; (ii) $u=r / a, \quad v=r^{2} \cos ^{2} \theta$; (iii) $u=r / a, \quad v=1 / r$.

Comment on your results in the light of part (a).

3.II.12C

commentLet $A$ be the closed planar region given by

$y \leqslant x \leqslant 2 y, \quad \frac{1}{y} \leqslant x \leqslant \frac{2}{y} .$

(i) Evaluate by means of a suitable change of variables the integral

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

(ii) Let $C$ be the boundary of $A$. Evaluate the line integral

$\oint_{C} \frac{x^{2}}{2 y} d y-d x$

by integrating along each section of the boundary.

(iii) Comment on your results.

3.II.9C

commentLet $\mathbf{F}=\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{x})$, where $\mathbf{x}$ is the position vector and $\boldsymbol{\omega}$ is a uniform vector field.

(i) Use the divergence theorem to evaluate the surface integral $\int_{S} \mathbf{F} \cdot d \mathbf{S}$, where $S$ is the closed surface of the cube with vertices $(\pm 1, \pm 1, \pm 1)$.

(ii) Show that $\boldsymbol{\nabla} \times \mathbf{F}=0$. Show further that the scalar field $\phi$ given by

$\phi=\frac{1}{2}(\boldsymbol{\omega} \cdot \mathbf{x})^{2}-\frac{1}{2}(\boldsymbol{\omega} \cdot \boldsymbol{\omega})(\mathbf{x} \cdot \mathbf{x})$

satisfies $\mathbf{F}=\boldsymbol{\nabla} \phi$. Describe geometrically the surfaces of constant $\phi$.