• # Paper 3, Section I, D

How many cyclic subgroups (including the trivial subgroup) does $S_{5}$ contain? Exhibit two isomorphic subgroups of $S_{5}$ which are not conjugate.

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• # Paper 3, Section I, D

Say that a group is dihedral if it has two generators $x$ and $y$, such that $x$ has order $n$ (greater than or equal to 2 and possibly infinite), $y$ has order 2 , and $y x y^{-1}=x^{-1}$. In particular the groups $C_{2}$ and $C_{2} \times C_{2}$ are regarded as dihedral groups. Prove that:

(i) any dihedral group can be generated by two elements of order 2 ;

(ii) any group generated by two elements of order 2 is dihedral; and

(iii) any non-trivial quotient group of a dihedral group is dihedral.

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• # Paper 3, Section II, D

(a) Let $G$ be a non-trivial group and let $Z(G)=\{h \in G: g h=h g$ for all $g \in G\}$. Show that $Z(G)$ is a normal subgroup of $G$. If the order of $G$ is a power of a prime, show that $Z(G)$ is non-trivial.

(b) The Heisenberg group $H$ is the set of all $3 \times 3$ matrices of the form

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

with $x, y, z \in \mathbb{R}$. Show that $H$ is a subgroup of the group of non-singular real matrices under matrix multiplication.

Find $Z(H)$ and show that $H / Z(H)$ is isomorphic to $\mathbb{R}^{2}$ under vector addition.

(c) For $p$ prime, the modular Heisenberg group $H_{p}$ is defined as in (b), except that $x, y$ and $z$ now lie in the field of $p$ elements. Write down $\left|H_{p}\right|$. Find both $Z\left(H_{p}\right)$ and $H_{p} / Z\left(H_{p}\right)$ in terms of generators and relations.

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• # Paper 3, Section II, D

(a) State and prove Lagrange's theorem.

(b) Let $G$ be a group and let $H, K$ be fixed subgroups of $G$. For each $g \in G$, any set of the form $H g K=\{h g k: h \in H, k \in K\}$ is called an $(H, K)$ double coset, or simply a double coset if $H$ and $K$ are understood. Prove that every element of $G$ lies in some $(H, K)$ double coset, and that any two $(H, K)$ double cosets either coincide or are disjoint.

Let $G$ be a finite group. Which of the following three statements are true, and which are false? Justify your answers.

(i) The size of a double coset divides the order of $G$.

(ii) Different double cosets for the same pair of subgroups have the same size.

(iii) The number of double cosets divides the order of $G$.

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• # Paper 3, Section II, D

Let $G, H$ be groups and let $\varphi: G \rightarrow H$ be a function. What does it mean to say that $\varphi$ is a homomorphism with kernel $K$ ? Show that if $K=\{e, \xi\}$ has order 2 then $x^{-1} \xi x=\xi$ for each $x \in G$. [If you use any general results about kernels of homomorphisms, then you should prove them.]

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

(a) There is a homomorphism from the orthogonal group $\mathrm{O}(3)$ to a group of order 2 with kernel the special orthogonal group $\mathrm{SO}(3)$.

(b) There is a homomorphism from the symmetry group $S_{3}$ of an equilateral triangle to a group of order 2 with kernel of order 3 .

(c) There is a homomorphism from $\mathrm{O}(3)$ to $\mathrm{SO}(3)$ with kernel of order 2 .

(d) There is a homomorphism from $S_{3}$ to a group of order 3 with kernel of order 2 .

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• # Paper 3, Section II, D

What does it mean for a group $G$ to act on a set $X$ ? For $x \in X$, what is meant by the orbit $\operatorname{Orb}(x)$ to which $x$ belongs, and by the stabiliser $G_{x}$ of $x$ ? Show that $G_{x}$ is a subgroup of $G$. Prove that, if $G$ is finite, then $|G|=\left|G_{x}\right| \cdot|\operatorname{Orb}(x)|$.

(a) Prove that the symmetric group $S_{n}$ acts on the set $P^{(n)}$ of all polynomials in $n$ variables $x_{1}, \ldots, x_{n}$, if we define $\sigma \cdot f$ to be the polynomial given by

$(\sigma \cdot f)\left(x_{1}, \ldots, x_{n}\right)=f\left(x_{\sigma(1)}, \ldots, x_{\sigma(n)}\right)$

for $f \in P^{(n)}$ and $\sigma \in S_{n}$. Find the orbit of $f=x_{1} x_{2}+x_{3} x_{4} \in P^{(4)}$ under $S_{4}$. Find also the order of the stabiliser of $f$.

(b) Let $r, n$ be fixed positive integers such that $r \leqslant n$. Let $B_{r}$ be the set of all subsets of size $r$ of the set $\{1,2, \ldots, n\}$. Show that $S_{n}$ acts on $B_{r}$ by defining $\sigma \cdot U$ to be the set $\{\sigma(u): u \in U\}$, for any $U \in B_{r}$ and $\sigma \in S_{n}$. Prove that $S_{n}$ is transitive in its action on $B_{r}$. Find also the size of the stabiliser of $U \in B_{r}$.

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• # Paper 3, Section I, A

The smooth curve $\mathcal{C}$ in $\mathbb{R}^{3}$ is given in parametrised form by the function $\mathbf{x}(u)$. Let $s$ denote arc length measured along the curve.

(a) Express the tangent $\mathbf{t}$ in terms of the derivative $\mathbf{x}^{\prime}=d \mathbf{x} / d u$, and show that $d u / d s=\left|\mathbf{x}^{\prime}\right|^{-1}$.

(b) Find an expression for $d \mathbf{t} / d s$ in terms of derivatives of $\mathbf{x}$ with respect to $u$, and show that the curvature $\kappa$ is given by

$\kappa=\frac{\left|\mathbf{x}^{\prime} \times \mathbf{x}^{\prime \prime}\right|}{\left|\mathbf{x}^{\prime}\right|^{3}}$

[Hint: You may find the identity $\left(\mathbf{x}^{\prime} \cdot \mathbf{x}^{\prime \prime}\right) \mathbf{x}^{\prime}-\left(\mathbf{x}^{\prime} \cdot \mathbf{x}^{\prime}\right) \mathbf{x}^{\prime \prime}=\mathbf{x}^{\prime} \times\left(\mathbf{x}^{\prime} \times \mathbf{x}^{\prime \prime}\right)$ helpful.]

(c) For the curve

$\mathbf{x}(u)=\left(\begin{array}{c} u \cos u \\ u \sin u \\ 0 \end{array}\right)$

with $u \geqslant 0$, find the curvature as a function of $u$.

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• # Paper 3, Section I, A

(i) For $r=|\mathbf{x}|$ with $\mathbf{x} \in \mathbb{R}^{3} \backslash\{\mathbf{0}\}$, show that

$\frac{\partial r}{\partial x_{i}}=\frac{x_{i}}{r} \quad(i=1,2,3) .$

(ii) Consider the vector fields $\mathbf{F}(\mathbf{x})=r^{2} \mathbf{x}, \mathbf{G}(\mathbf{x})=(\mathbf{a} \cdot \mathbf{x}) \mathbf{x}$ and $\mathbf{H}(\mathbf{x})=\mathbf{a} \times \hat{\mathbf{x}}$, where $\mathbf{a}$ is a constant vector in $\mathbb{R}^{3}$ and $\hat{\mathbf{x}}$ is the unit vector in the direction of $\mathbf{x}$. Using suffix notation, or otherwise, find the divergence and the curl of each of $\mathbf{F}, \mathbf{G}$ and $\mathbf{H}$.

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• # Paper 3, Section II, A

(a) Let $t_{i j}$ be a rank 2 tensor whose components are invariant under rotations through an angle $\pi$ about each of the three coordinate axes. Show that $t_{i j}$ is diagonal.

(b) An array of numbers $a_{i j}$ is given in one orthonormal basis as $\delta_{i j}+\epsilon_{1 i j}$ and in another rotated basis as $\delta_{i j}$. By using the invariance of the determinant of any rank 2 tensor, or otherwise, prove that $a_{i j}$ is not a tensor.

(c) Let $a_{i j}$ be an array of numbers and $b_{i j}$ a tensor. Determine whether the following statements are true or false. Justify your answers.

(i) If $a_{i j} b_{i j}$ is a scalar for any rank 2 tensor $b_{i j}$, then $a_{i j}$ is a rank 2 tensor.

(ii) If $a_{i j} b_{i j}$ is a scalar for any symmetric rank 2 tensor $b_{i j}$, then $a_{i j}$ is a rank 2 tensor.

(iii) If $a_{i j}$ is antisymmetric and $a_{i j} b_{i j}$ is a scalar for any symmetric rank 2 tensor $b_{i j}$, then $a_{i j}$ is an antisymmetric rank 2 tensor.

(iv) If $a_{i j}$ is antisymmetric and $a_{i j} b_{i j}$ is a scalar for any antisymmetric rank 2 tensor $b_{i j}$, then $a_{i j}$ is an antisymmetric rank 2 tensor.

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• # Paper 3, Section II, A

(i) Starting with the divergence theorem, derive Green's first theorem

$\int_{V}\left(\psi \nabla^{2} \phi+\nabla \psi \cdot \nabla \phi\right) d V=\int_{\partial V} \psi \frac{\partial \phi}{\partial n} d S$

(ii) The function $\phi(\mathbf{x})$ satisfies Laplace's equation $\nabla^{2} \phi=0$ in the volume $V$ with given boundary conditions $\phi(\mathbf{x})=g(\mathbf{x})$ for all $\mathbf{x} \in \partial V$. Show that $\phi(\mathbf{x})$ is the only such function. Deduce that if $\phi(\mathbf{x})$ is constant on $\partial V$ then it is constant in the whole volume $V$.

(iii) Suppose that $\phi(\mathbf{x})$ satisfies Laplace's equation in the volume $V$. Let $V_{r}$ be the sphere of radius $r$ centred at the origin and contained in $V$. The function $f(r)$ is defined by

$f(r)=\frac{1}{4 \pi r^{2}} \int_{\partial V_{r}} \phi(\mathbf{x}) d S$

By considering the derivative $d f / d r$, and by introducing the Jacobian in spherical polar coordinates and using the divergence theorem, or otherwise, show that $f(r)$ is constant and that $f(r)=\phi(\mathbf{0})$.

(iv) Let $M$ denote the maximum of $\phi$ on $\partial V_{r}$ and $m$ the minimum of $\phi$ on $\partial V_{r}$. By using the result from (iii), or otherwise, show that $m \leqslant \phi(\mathbf{0}) \leqslant M$.

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• # Paper 3, Section II, A

State Stokes' theorem.

Let $S$ be the surface in $\mathbb{R}^{3}$ given by $z^{2}=x^{2}+y^{2}+1-\lambda$, where $0 \leqslant z \leqslant 1$ and $\lambda$ is a positive constant. Sketch the surface $S$ for representative values of $\lambda$ and find the surface element $\mathbf{d} \mathbf{S}$ with respect to the Cartesian coordinates $x$ and $y$.

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

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

and verify Stokes' theorem for $\mathbf{F}$ on the surface $S$ for every value of $\lambda$.

Now compute $\nabla \times \mathbf{G}$ for the vector field

$\mathbf{G}(\mathbf{x})=\frac{1}{x^{2}+y^{2}}\left(\begin{array}{c} -y \\ x \\ 0 \end{array}\right)$

and find the line integral $\int_{\partial S} \mathbf{G} \cdot \mathbf{d x}$ for the boundary $\partial S$ of the surface $S$. Is it possible to obtain this result using Stokes' theorem? Justify your answer.

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• # Paper 3, Section II, A

The vector field $\mathbf{F}(\mathbf{x})$ is given in terms of cylindrical polar coordinates $(\rho, \phi, z)$ by

$\mathbf{F}(\mathbf{x})=f(\rho) \mathbf{e}_{\rho}$

where $f$ is a differentiable function of $\rho$, and $\mathbf{e}_{\rho}=\cos \phi \mathbf{e}_{x}+\sin \phi \mathbf{e}_{y}$ is the unit basis vector with respect to the coordinate $\rho$. Compute the partial derivatives $\partial F_{1} / \partial x, \partial F_{2} / \partial y$, $\partial F_{3} / \partial z$ and hence find the divergence $\nabla \cdot \mathbf{F}$ in terms of $\rho$ and $\phi$.

The domain $V$ is bounded by the surface $z=\left(x^{2}+y^{2}\right)^{-1}$, by the cylinder $x^{2}+y^{2}=1$, and by the planes $z=\frac{1}{4}$ and $z=1$. Sketch $V$ and compute its volume.

Find the most general function $f(\rho)$ such that $\nabla \cdot \mathbf{F}=0$, and verify the divergence theorem for the corresponding vector field $\mathbf{F}(\mathbf{x})$ in $V$.

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