Part IA, 2013, Paper 3

# Part IA, 2013, Paper 3

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Paper 3, Section I, D

commentDefine what it means for a group to be cyclic, and for a group to be abelian. Show that every cyclic group is abelian, and give an example to show that the converse is false.

Show that a group homomorphism from the cyclic group $C_{n}$ of order $n$ to a group $G$ determines, and is determined by, an element $g$ of $G$ such that $g^{n}=1$.

Hence list all group homomorphisms from $C_{4}$ to the symmetric group $S_{4}$.

Paper 3, Section I, D

commentState Lagrange's Theorem.

Let $G$ be a finite group, and $H$ and $K$ two subgroups of $G$ such that

(i) the orders of $H$ and $K$ are coprime;

(ii) every element of $G$ may be written as a product $h k$, with $h \in H$ and $k \in K$;

(iii) both $H$ and $K$ are normal subgroups of $G$.

Prove that $G$ is isomorphic to $H \times K$.

Paper 3, Section II, D

commentLet $p$ be a prime number.

Prove that every group whose order is a power of $p$ has a non-trivial centre.

Show that every group of order $p^{2}$ is abelian, and that there are precisely two of them, up to isomorphism.

Paper 3, Section II, D

comment(a) Let $G$ be the dihedral group of order $4 n$, the symmetry group of a regular polygon with $2 n$ sides.

Determine all elements of order 2 in $G$. For each element of order 2 , determine its conjugacy class and the smallest normal subgroup containing it.

(b) Let $G$ be a finite group.

(i) Prove that if $H$ and $K$ are subgroups of $G$, then $K \cup H$ is a subgroup if and only if $H \subseteq K$ or $K \subseteq H$.

(ii) Let $H$ be a proper subgroup of $G$, and write $G \backslash H$ for the elements of $G$ not in $H$. Let $K$ be the subgroup of $G$ generated by $G \backslash H$.

Show that $K=G$.

Paper 3, Section II, D

comment(a) Let $p$ be a prime, and let $G=S L_{2}(p)$ be the group of $2 \times 2$ matrices of determinant 1 with entries in the field $\mathbb{F}_{p}$ of integers $\bmod p$.

(i) Define the action of $G$ on $X=\mathbb{F}_{p} \cup\{\infty\}$ by MÃ¶bius transformations. [You need not show that it is a group action.]

State the orbit-stabiliser theorem.

Determine the orbit of $\infty$ and the stabiliser of $\infty$. Hence compute the order of $S L_{2}(p)$.

(ii) Let

$A=\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right), \quad B=\left(\begin{array}{ll} 1 & 3 \\ 0 & 1 \end{array}\right)$

Show that $A$ is conjugate to $B$ in $G$ if $p=11$, but not if $p=5$.

(b) Let $G$ be the set of all $3 \times 3$ matrices of the form

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

where $a, b, x \in \mathbb{R}$. Show that $G$ is a subgroup of the group of all invertible real matrices.

Let $H$ be the subset of $G$ given by matrices with $a=0$. Show that $H$ is a normal subgroup, and that the quotient group $G / H$ is isomorphic to $\mathbb{R}$.

Determine the centre $Z(G)$ of $G$, and identify the quotient group $G / Z(G)$.

Paper 3, Section II, D

comment(a) Let $G$ be a finite group. Show that there exists an injective homomorphism $G \rightarrow \operatorname{Sym}(X)$ to a symmetric group, for some set $X$.

(b) Let $H$ be the full group of symmetries of the cube, and $X$ the set of edges of the cube.

Show that $H$ acts transitively on $X$, and determine the stabiliser of an element of $X$. Hence determine the order of $H$.

Show that the action of $H$ on $X$ defines an injective homomorphism $H \rightarrow \operatorname{Sym}(X)$ to the group of permutations of $X$, and determine the number of cosets of $H$ in $\operatorname{Sym}(X)$.

Is $H$ a normal subgroup of $\operatorname{Sym}(X) ?$ Prove your answer.

Paper 3, Section $I$, C

commentState a necessary and sufficient condition for a vector field $\mathbf{F}$ on $\mathbb{R}^{3}$ to be conservative.

Check that the field

$\mathbf{F}=\left(2 x \cos y-2 z^{3}, 3+2 y e^{z}-x^{2} \sin y, y^{2} e^{z}-6 x z^{2}\right)$

is conservative and find a scalar potential for $\mathbf{F}$.

Paper 3, Section I, C

commentThe curve $C$ is given by

$\mathbf{r}(t)=\left(\sqrt{2} e^{t},-e^{t} \sin t, e^{t} \cos t\right), \quad-\infty<t<\infty$

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

(ii) Derive an expression for the curvature of $C$ as a function of arc length $s$ measured from the point with $t=0$.

Paper 3, Section II, C

comment(a) Prove that

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

(b) State the divergence theorem for a vector field $\mathbf{F}$ in a closed region $\Omega \subset \mathbb{R}^{3}$ bounded by $\partial \Omega$.

For a smooth vector field $\mathbf{F}$ and a smooth scalar function $g$ prove that

$\int_{\Omega} \mathbf{F} \cdot \nabla g+g \nabla \cdot \mathbf{F} d V=\int_{\partial \Omega} g \mathbf{F} \cdot \mathbf{n} d S,$

where $\mathbf{n}$ is the outward unit normal on the surface $\partial \Omega$.

Use this identity to prove that the solution $u$ to the Laplace equation $\nabla^{2} u=0$ in $\Omega$ with $u=f$ on $\partial \Omega$ is unique, provided it exists.

Paper 3, Section II, C

commentIf $\mathbf{E}$ and $\mathbf{B}$ are vectors in $\mathbb{R}^{3}$, show that

$T_{i j}=E_{i} E_{j}+B_{i} B_{j}-\frac{1}{2} \delta_{i j}\left(E_{k} E_{k}+B_{k} B_{k}\right)$

is a second rank tensor.

Now assume that $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ obey Maxwell's equations, which in suitable units read

$\begin{aligned} &\nabla \cdot \mathbf{E}=\rho \\ &\nabla \cdot \mathbf{B}=0 \\ &\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla \times \mathbf{B}=\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t} \end{aligned}$

where $\rho$ is the charge density and $\mathbf{J}$ the current density. Show that

$\frac{\partial}{\partial t}(\mathbf{E} \times \mathbf{B})=\mathbf{M}-\rho \mathbf{E}-\mathbf{J} \times \mathbf{B} \quad \text { where } \quad M_{i}=\frac{\partial T_{i j}}{\partial x_{j}}$

Paper 3, Section II, C

commentConsider the bounded surface $S$ that is the union of $x^{2}+y^{2}=4$ for $-2 \leqslant z \leqslant 2$ and $(4-z)^{2}=x^{2}+y^{2}$ for $2 \leqslant z \leqslant 4$. Sketch the surface.

Using suitable parametrisations for the two parts of $S$, calculate the integral

$\int_{S}(\nabla \times \mathbf{F}) \cdot d \mathbf{S}$

for $\mathbf{F}=y z^{2} \mathbf{i}$.

Check your result using Stokes's Theorem.

Paper 3, Section II, C

commentGive an explicit formula for $\mathcal{J}$ which makes the following result hold:

$\int_{D} f d x d y d z=\int_{D^{\prime}} \phi|\mathcal{J}| d u d v d w$

where the region $D$, with coordinates $x, y, z$, and the region $D^{\prime}$, with coordinates $u, v, w$, are in one-to-one correspondence, and

$\phi(u, v, w)=f(x(u, v, w), y(u, v, w), z(u, v, w))$

Explain, in outline, why this result holds.

Let $D$ be the region in $\mathbb{R}^{3}$ defined by $4 \leqslant x^{2}+y^{2}+z^{2} \leqslant 9$ and $z \geqslant 0$. Sketch the region and employ a suitable transformation to evaluate the integral

$\int_{D}\left(x^{2}+y^{2}\right) d x d y d z$