Part IA, 2012, Paper 3

Groups

• Paper 3, Section I, E

  Groups | Part IA, 2012

  What 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.

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• Paper 3, Section I, E

  Groups | Part IA, 2012

  State 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 $g^{4}=e$.

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• Paper 3, Section II, $7 \mathrm{E}$

  Groups | Part IA, 2012

  Let $\mathbb{F}_{p}$ 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

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

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• Paper 3, Section II, E

  Groups | Part IA, 2012

  Let $p$ 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 $K$.

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• Paper 3, Section II, E

  Groups | Part IA, 2012

  Let $G$ be $S L_{2}(\mathbb{R})$, the groups of real $2 \times 2$ matrices of determinant 1 , acting on $\mathbb{C} \cup\{\infty\}$ by Möbius 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?

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• Paper 3, Section II, E

  Groups | Part IA, 2012

  (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 $S_{n}$.

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Vector Calculus

• Paper 3, Section I, C

  Vector Calculus | Part IA, 2012

  What 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 $\lambda$.

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• Paper 3, Section I, C

  Vector Calculus | Part IA, 2012

  Define 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.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2012

  (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).

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2012

  (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 $\phi$.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2012

  Consider 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

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

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2012

  State 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.

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