Part IA, 2012, Paper 3

# Part IA, 2012, Paper 3

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Paper 3, Section I, E

commentWhat is a cycle in the symmetric group $S_{n}$ ? Show that a cycle of length $p$ and a cycle of length $q$ in $S_{n}$ are conjugate if and only if $p=q$.

Suppose that $p$ is odd. Show that any two $p$-cycles in $A_{p+2}$ are conjugate. Are any two 3 -cycles in $A_{4}$ conjugate? Justify your answer.

Paper 3, Section I, E

commentState Lagrange's Theorem. Deduce that if $G$ is a finite group of order $n$, then the order of every element of $G$ is a divisor of $n$.

Let $G$ be a group such that, for every $g \in G, g^{2}=e$. Show that $G$ is abelian. Give an example of a non-abelian group in which every element $g$ satisfies $g^{4}=e$.

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

commentLet $\mathbb{F}_{p}$ be the set of (residue classes of) integers $\bmod p$, and let

$G=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right): a, b, c, d \in \mathbb{F}_{p}, a d-b c \neq 0\right\}$

Show that $G$ is a group under multiplication. [You may assume throughout this question that multiplication of matrices is associative.]

Let $X$ be the set of 2-dimensional column vectors with entries in $\mathbb{F}_{p}$. Show that the mapping $G \times X \rightarrow X$ given by

$\left(\left(\begin{array}{ll} a & b \\ c & d \end{array}\right),\left(\begin{array}{l} x \\ y \end{array}\right)\right) \mapsto\left(\begin{array}{l} a x+b y \\ c x+d y \end{array}\right)$

is a group action.

Let $g \in G$ be an element of order $p$. Use the orbit-stabilizer theorem to show that there exist $x, y \in \mathbb{F}_{p}$, not both zero, with

$g\left(\begin{array}{l} x \\ y \end{array}\right)=\left(\begin{array}{l} x \\ y \end{array}\right)$

Deduce that $g$ is conjugate in $G$ to the matrix

$\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right)$

Paper 3, Section II, E

commentLet $p$ be a prime number, and $a$ an integer with $1 \leqslant a \leqslant p-1$. Let $G$ be the Cartesian product

$G=\{(x, u) \mid x \in\{0,1, \ldots, p-2\}, u \in\{0,1, \ldots, p-1\}\}$

Show that the binary operation

$(x, u) *(y, v)=(z, w)$

where

$\begin{aligned} z & \equiv x+y(\bmod p-1) \\ w & \equiv a^{y} u+v(\bmod p) \end{aligned}$

makes $G$ into a group. Show that $G$ is abelian if and only if $a=1$.

Let $H$ and $K$ be the subsets

$H=\{(x, 0) \mid x \in\{0,1, \ldots, p-2\}\}, \quad K=\{(0, u) \mid u \in\{0,1, \ldots, p-1\}\}$

of $G$. Show that $K$ is a normal subgroup of $G$, and that $H$ is a subgroup which is normal if and only if $a=1$.

Find a homomorphism from $G$ to another group whose kernel is $K$.

Paper 3, Section II, E

commentLet $G$ be $S L_{2}(\mathbb{R})$, the groups of real $2 \times 2$ matrices of determinant 1 , acting on $\mathbb{C} \cup\{\infty\}$ by MÃ¶bius transformations.

For each of the points $0, i,-i$, compute its stabilizer and its orbit under the action of $G$. Show that $G$ has exactly 3 orbits in all.

Compute the orbit of $i$ under the subgroup

$H=\left\{\left(\begin{array}{ll} a & b \\ 0 & d \end{array}\right) \mid a, b, d \in \mathbb{R}, a d=1\right\} \subset G .$

Deduce that every element $g$ of $G$ may be expressed in the form $g=h k$ where $h \in H$ and for some $\theta \in \mathbb{R}$,

$k=\left(\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$

How many ways are there of writing $g$ in this form?

Paper 3, Section II, E

comment(i) State and prove the Orbit-Stabilizer Theorem.

Show that if $G$ is a finite group of order $n$, then $G$ is isomorphic to a subgroup of the symmetric group $S_{n}$.

(ii) Let $G$ be a group acting on a set $X$ with a single orbit, and let $H$ be the stabilizer of some element of $X$. Show that the homomorphism $G \rightarrow \operatorname{Sym}(X)$ given by the action is injective if and only if the intersection of all the conjugates of $H$ equals $\{e\}$.

(iii) Let $Q_{8}$ denote the quaternion group of order 8 . Show that for every $n<8, Q_{8}$ is not isomorphic to a subgroup of $S_{n}$.

Paper 3, Section I, C

commentWhat does it mean for a second-rank tensor $T_{i j}$ to be isotropic? Show that $\delta_{i j}$ is isotropic. By considering rotations through $\pi / 2$ about the coordinate axes, or otherwise, show that the most general isotropic second-rank tensor in $\mathbb{R}^{3}$ has the form $T_{i j}=\lambda \delta_{i j}$, for some scalar $\lambda$.

Paper 3, Section I, C

commentDefine what it means for a differential $P d x+Q d y$ to be exact, and derive a necessary condition on $P(x, y)$ and $Q(x, y)$ for this to hold. Show that one of the following two differentials is exact and the other is not:

$\begin{aligned} &y^{2} d x+2 x y d y \\ &y^{2} d x+x y^{2} d y \end{aligned}$

Show that the differential which is not exact can be written in the form $g d f$ for functions $f(x, y)$ and $g(y)$, to be determined.

Paper 3, Section II, C

comment(i) Let $V$ be a bounded region in $\mathbb{R}^{3}$ with smooth boundary $S=\partial V$. Show that Poisson's equation in $V$

$\nabla^{2} u=\rho$

has at most one solution satisfying $u=f$ on $S$, where $\rho$ and $f$ are given functions.

Consider the alternative boundary condition $\partial u / \partial n=g$ on $S$, for some given function $g$, where $n$ is the outward pointing normal on $S$. Derive a necessary condition in terms of $\rho$ and $g$ for a solution $u$ of Poisson's equation to exist. Is such a solution unique?

(ii) Find the most general spherically symmetric function $u(r)$ satisfying

$\nabla^{2} u=1$

in the region $r=|\mathbf{r}| \leqslant a$ for $a>0$. Hence in each of the following cases find all possible solutions satisfying the given boundary condition at $r=a$ : (a) $u=0$, (b) $\frac{\partial u}{\partial n}=0$.

Compare these with your results in part (i).

Paper 3, Section II, C

comment(a) Prove the identity

$\nabla(\mathbf{F} \cdot \mathbf{G})=(\mathbf{F} \cdot \nabla) \mathbf{G}+(\mathbf{G} \cdot \nabla) \mathbf{F}+\mathbf{F} \times(\nabla \times \mathbf{G})+\mathbf{G} \times(\nabla \times \mathbf{F})$

(b) If $\mathbf{E}$ is an irrotational vector field (i.e. $\nabla \times \mathbf{E}=\mathbf{0}$ everywhere), prove that there exists a scalar potential $\phi(\mathbf{x})$ such that $\mathbf{E}=-\nabla \phi$.

Show that the vector field

$\left(x y^{2} z e^{-x^{2} z},-y e^{-x^{2} z}, \frac{1}{2} x^{2} y^{2} e^{-x^{2} z}\right)$

is irrotational, and determine the corresponding potential $\phi$.

Paper 3, Section II, C

commentConsider the transformation of variables

$x=1-u, \quad y=\frac{1-v}{1-u v} .$

Show that the interior of the unit square in the $u v$ plane

$\{(u, v): 0<u<1,0<v<1\}$

is mapped to the interior of the unit square in the $x y$ plane,

$R=\{(x, y): 0<x<1,0<y<1\} .$

[Hint: Consider the relation between $v$ and $y$ when $u=\alpha$, for $0<\alpha<1$ constant.]

Show that

$\frac{\partial(x, y)}{\partial(u, v)}=\frac{(1-(1-x) y)^{2}}{x}$

Now let

$u=\frac{1-t}{1-w t}, \quad v=1-w$

By calculating

$\frac{\partial(x, y)}{\partial(t, w)}=\frac{\partial(x, y)}{\partial(u, v)} \frac{\partial(u, v)}{\partial(t, w)}$

as a function of $x$ and $y$, or otherwise, show that

$\int_{R} \frac{x(1-y)}{(1-(1-x) y)\left(1-\left(1-x^{2}\right) y\right)^{2}} d x d y=1$

Paper 3, Section II, C

commentState Stokes' Theorem for a vector field $\mathbf{B}(\mathbf{x})$ on $\mathbb{R}^{3}$.

Consider the surface $S$ defined by

$z=x^{2}+y^{2}, \quad \frac{1}{9} \leqslant z \leqslant 1 .$

Sketch the surface and calculate the area element $d \mathbf{S}$ in terms of suitable coordinates or parameters. For the vector field

$\mathbf{B}=\left(-y^{3}, x^{3}, z^{3}\right)$

compute $\nabla \times \mathbf{B}$ and calculate $I=\int_{S}(\nabla \times \mathbf{B}) \cdot d \mathbf{S}$.

Use Stokes' Theorem to express $I$ as an integral over $\partial S$ and verify that this gives the same result.