# Groups

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Paper 3 , Section I, D

commentLet $G$ be a finite group and denote the centre of $G$ by $Z(G)$. Prove that if the quotient group $G / Z(G)$ is cyclic then $G$ is abelian. Does there exist a group $H$ such that (i) $|H / Z(H)|=7$ ? (ii) $|H| Z(H) \mid=6$ ?

Justify your answers.

Paper 3, Section I, D

commentLet $g$ and $h$ be elements of a group $G$. What does it mean to say $g$ and $h$ are conjugate in $G$ ? Prove that if two elements in a group are conjugate then they have the same order.

Define the MÃ¶bius group $\mathcal{M}$. Prove that if $g, h \in \mathcal{M}$ are conjugate they have the same number of fixed points. Quoting clearly any results you use, show that any nontrivial element of $\mathcal{M}$ of finite order has precisely 2 fixed points.

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

commentLet $G$ be a finite group of order $n$. Show that $G$ is isomorphic to a subgroup $H$ of $S_{n}$, the symmetric group of degree $n$. Furthermore show that this isomorphism can be chosen so that any nontrivial element of $H$ has no fixed points.

Suppose $n$ is even. Prove that $G$ contains an element of order 2 .

What does it mean for an element of $S_{m}$ to be odd? Suppose $H$ is a subgroup of $S_{m}$ for some $m$, and $H$ contains an odd element. Prove that precisely half of the elements of $H$ are odd.

Now suppose $n=4 k+2$ for some positive integer $k$. Prove that $G$ is not simple. [Hint: Consider the sign of an element of order 2.]

Can a nonabelian group of even order be simple?

Paper 3, Section II, D

comment(a) Let $A$ be an abelian group (not necessarily finite). We define the generalised dihedral group to be the set of pairs

$D(A)=\{(a, \varepsilon): a \in A, \varepsilon=\pm 1\}$

with multiplication given by

$(a, \varepsilon)(b, \eta)=\left(a b^{\varepsilon}, \varepsilon \eta\right)$

The identity is $(e, 1)$ and the inverse of $(a, \varepsilon)$ is $\left(a^{-\varepsilon}, \varepsilon\right)$. You may assume that this multiplication defines a group operation on $D(A)$.

(i) Identify $A$ with the set of all pairs in which $\varepsilon=+1$. Show that $A$ is a subgroup of $D(A)$. By considering the index of $A$ in $D(A)$, or otherwise, show that $A$ is a normal subgroup of $D(A)$.

(ii) Show that every element of $D(A)$ not in $A$ has order 2 . Show that $D(A)$ is abelian if and only if $a^{2}=e$ for all $a \in A$. If $D(A)$ is non-abelian, what is the centre of $D(A) ?$ Justify your answer.

(b) Let $\mathrm{O}(2)$ denote the group of $2 \times 2$ orthogonal matrices. Show that all elements of $\mathrm{O}(2)$ have determinant 1 or $-1$. Show that every element of $\mathrm{SO}(2)$ is a rotation. Let $J=\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right)$. Show that $\mathrm{O}(2)$ decomposes as a union $\mathrm{SO}(2) \cup \operatorname{SO}(2) J$.

[You may assume standard properties of determinants.]

(c) Let $B$ be the (abelian) group $\{z \in \mathbb{C}:|z|=1\}$, with multiplication of complex numbers as the group operation. Write down, without proof, isomorphisms $\mathrm{SO}(2) \cong B \cong \mathbb{R} / \mathbb{Z}$ where $\mathbb{R}$ denotes the additive group of real numbers and $\mathbb{Z}$ the subgroup of integers. Deduce that $\mathrm{O}(2) \cong D(B)$, the generalised dihedral group defined in part (a).

Paper 3, Section II, D

comment(a) Let $G$ be a finite group acting on a set $X$. For $x \in X$, define the orbit $\operatorname{Orb}(x)$ and the stabiliser $\operatorname{Stab}(x)$ of $x$. Show that $\operatorname{Stab}(x)$ is a subgroup of $G$. State and prove the orbit-stabiliser theorem.

(b) Let $n \geqslant k \geqslant 1$ be integers. Let $G=S_{n}$, the symmetric group of degree $n$, and $X$ be the set of all ordered $k$-tuples $\left(x_{1}, \ldots, x_{k}\right)$ with $x_{i} \in\{1,2, \ldots, n\}$. Then $G$ acts on $X$, where the action is defined by $\sigma\left(x_{1}, \ldots, x_{k}\right)=\left(\sigma\left(x_{1}\right), \ldots, \sigma\left(x_{k}\right)\right)$ for $\sigma \in S_{n}$ and $\left(x_{1}, \ldots, x_{k}\right) \in X$. For $x=(1,2, \ldots, k) \in X$, determine $\operatorname{Orb}(x)$ and $\operatorname{Stab}(x)$ and verify that the orbit-stabiliser theorem holds in this case.

(c) We say that $G$ acts doubly transitively on $X$ if, whenever $\left(x_{1}, x_{2}\right)$ and $\left(y_{1}, y_{2}\right)$ are elements of $X \times X$ with $x_{1} \neq x_{2}$ and $y_{1} \neq y_{2}$, there exists some $g \in G$ such that $g x_{1}=y_{1}$ and $g x_{2}=y_{2}$.

Assume that $G$ is a finite group that acts doubly transitively on $X$, and let $x \in X$. Show that if $H$ is a subgroup of $G$ that properly contains $\operatorname{Stab}(x)($ that is, $\operatorname{Stab}(x) \subseteq H$ but $\operatorname{Stab}(x) \neq H)$ then the action of $H$ on $X$ is transitive. Deduce that $H=G$.

Paper 3, Section II, D

comment(a) Let $x$ be an element of a finite group $G$. Define the order of $x$ and the order of $G$. State and prove Lagrange's theorem. Deduce that the order of $x$ divides the order of $G$.

(b) If $G$ is a group of order $n$, and $d$ is a divisor of $n$ where $d<n$, is it always true that $G$ must contain an element of order $d$ ? Justify your answer.

(c) Denote the cyclic group of order $m$ by $C_{m}$.

(i) Prove that if $m$ and $n$ are coprime then the direct product $C_{m} \times C_{n}$ is cyclic.

(ii) Show that if a finite group $G$ has all non-identity elements of order 2 , then $G$ is isomorphic to $C_{2} \times \cdots \times C_{2}$. [The direct product theorem may be used without proof.]

(d) Let $G$ be a finite group and $H$ a subgroup of $G$.

(i) Let $x$ be an element of order $d$ in $G$. If $r$ is the least positive integer such that $x^{r} \in H$, show that $r$ divides $d$.

(ii) Suppose further that $H$ has index $n$. If $x \in G$, show that $x^{k} \in H$ for some $k$ such that $0<k \leqslant n$. Is it always the case that the least positive such $k$ is a factor of $n$ ? Justify your answer.

Paper 2, Section I, E

commentWhat does it mean for an element of the symmetric group $S_{n}$ to be a transposition or a cycle?

Let $n \geqslant 4$. How many permutations $\sigma$ of $\{1,2, \ldots, n\}$ are there such that

(i) $\sigma(1)=2 ?$

(ii) $\sigma(k)$ is even for each even number $k$ ?

(iii) $\sigma$ is a 4-cycle?

(iv) $\sigma$ can be written as the product of two transpositions?

You should indicate in each case how you have derived your formula.

Paper 2, Section II, E

comment(a) Let $G$ be a finite group acting on a finite set $X$. For any subset $T$ of $G$, we define the fixed point set as $X^{T}=\{x \in X: \forall g \in T, g \cdot x=x\}$. Write $X^{g}$ for $X^{\{g\}}(g \in G)$. Let $G \backslash X$ be the set of $G$-orbits in $X$. In what follows you may assume the orbit-stabiliser theorem.

Prove that

$|X|=\left|X^{G}\right|+\sum_{x}|G| /\left|G_{x}\right|$

where the sum is taken over a set of representatives for the orbits containing more than one element.

By considering the set $Z=\{(g, x) \in G \times X: g \cdot x=x\}$, or otherwise, show also that

$|G \backslash X|=\frac{1}{|G|} \sum_{g \in G}\left|X^{g}\right|$

(b) Let $V$ be the set of vertices of a regular pentagon and let the dihedral group $D_{10}$ act on $V$. Consider the set $X_{n}$ of functions $F: V \rightarrow \mathbb{Z}_{n}$ (the integers mod $\left.n\right)$. Assume that $D_{10}$ and its rotation subgroup $C_{5}$ act on $X_{n}$ by the rule

$(g \cdot F)(v)=F\left(g^{-1} \cdot v\right),$

where $g \in D_{10}, F \in X_{n}$ and $v \in V$. It is given that $\left|X_{n}\right|=n^{5}$. We define a necklace to be a $C_{5}$-orbit in $X_{n}$ and a bracelet to be a $D_{10}$-orbit in $X_{n}$.

Find the number of necklaces and bracelets for any $n$.

Paper 2, Section II, E

commentSuppose that $f$ is a Möbius transformation acting on the extended complex plane. Show that a Möbius transformation with at least three fixed points is the identity. Deduce that every Möbius transformation except the identity has one or two fixed points.

Which of the following statements are true and which are false? Justify your answers, quoting standard facts if required.

(i) If $f$ has exactly one fixed point then it is conjugate to $z \mapsto z+1$.

(ii) Every Möbius transformation 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).

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

(iv) The operation of complex conjugation defined by $z \mapsto \bar{z}$ is a Möbius transformation.

Paper 3, Section $I$, D

commentWhat is the orthogonal group $\mathrm{O}(n)$ ? What is the special orthogonal group $\mathrm{SO}(n) ?$

Show that every element of $\mathrm{SO}(3)$ has an eigenvector with eigenvalue $1 .$

Is it true that every element of $\mathrm{O}(3)$ is either a rotation or a reflection? Justify your answer.

Paper 3, Section I, D

commentProve that two elements of $S_{n}$ are conjugate if and only if they have the same cycle type.

Describe a condition on the centraliser (in $S_{n}$ ) of a permutation $\sigma \in A_{n}$ that ensures the conjugacy class of $\sigma$ in $A_{n}$ is the same as the conjugacy class of $\sigma$ in $S_{n}$. Justify your answer.

How many distinct conjugacy classes are there in $A_{5}$ ?

Paper 3, Section II, D

commentLet $\mathcal{M}$ be the group of Möbius transformations of $\mathbb{C} \cup\{\infty\}$ and let $\mathrm{SL}_{2}(\mathbb{C})$ be the group of all $2 \times 2$ complex matrices of determinant 1 .

Show that the map $\theta: \mathrm{SL}_{2}(\mathbb{C}) \rightarrow \mathcal{M}$ given by

$\theta\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)(z)=\frac{a z+b}{c z+d}$

is a surjective homomorphism. Find its kernel.

Show that any $T \in \mathcal{M}$ not equal to the identity is conjugate to a Möbius map $S$ where either $S z=\mu z$ with $\mu \neq 0,1$ or $S z=z+1$. [You may use results about matrices in $\mathrm{SL}_{2}(\mathbb{C})$ as long as they are clearly stated.]

Show that any non-identity Möbius map has one or two fixed points. Also show that if $T$ is a Möbius map with just one fixed point $z_{0}$ then $T^{n} z \rightarrow z_{0}$ as $n \rightarrow \infty$ for any $z \in \mathbb{C} \cup\{\infty\}$. [You may assume that Möbius maps are continuous.]

Paper 3, Section II, D

commentState and prove the first isomorphism theorem. [You may assume that kernels of homomorphisms are normal subgroups and images are subgroups.]

Let $G$ be a group with subgroup $H$ and normal subgroup $N$. Prove that $N H=\{n h: n \in N, h \in H\}$ is a subgroup of $G$ and $N \cap H$ is a normal subgroup of $H$. Further, show that $N$ is a normal subgroup of $N H$.

Prove that $\frac{H}{N \cap H}$ is isomorphic to $\frac{N H}{N}$.

If $K$ and $H$ are both normal subgroups of $G$ must $K H$ be a normal subgroup of $G$ ?

If $K$ and $H$ are subgroups of $G$, but not normal subgroups, must $K H$ be a subgroup of $G$ ?

Justify your answers.

Paper 3, Section II, D

commentState and prove Lagrange's Theorem.

Hence show that if $G$ is a finite group and $g \in G$ then the order of $g$ divides the order of $G$.

How many elements are there of order 3 in the following groups? Justify your answers.

(a) $C_{3} \times C_{9}$, where $C_{n}$ denotes the cyclic group of order $n$.

(b) $D_{2 n}$ the dihedral group of order $2 n$.

(c) $S_{7}$ the symmetric group of degree 7 .

(d) $A_{7}$ the alternating group of degree 7 .

Paper 3, Section II, D

commentLet $H$ and $K$ be subgroups of a group $G$ satisfying the following two properties.

(i) All elements of $G$ can be written in the form $h k$ for some $h \in H$ and some $k \in K$.

(ii) $H \cap K=\{e\}$.

Prove that $H$ and $K$ are normal subgroups of $G$ if and only if all elements of $H$ commute with all elements of $K$.

State and prove Cauchy's Theorem.

Let $p$ and $q$ be distinct primes. Prove that an abelian group of order $p q$ is isomorphic to $C_{p} \times C_{q}$. Is it true that all abelian groups of order $p^{2}$ are isomorphic to $C_{p} \times C_{p}$ ?

Paper 3, Section I, D

commentProve that every member of $O(3)$ is a product of at most three reflections.

Is every member of $O(3)$ a product of at most two reflections? Justify your answer.

Paper 3, Section I, D

commentFind the order and the sign of the permutation $(13)(2457)(815) \in S_{8}$.

How many elements of $S_{6}$ have order $6 ?$ And how many have order $3 ?$

What is the greatest order of any element of $A_{9}$ ?

Paper 3, Section II, D

commentState and prove the Direct Product Theorem.

Is the group $O(3)$ isomorphic to $S O(3) \times C_{2} ?$ Is $O(2)$ isomorphic to $S O(2) \times C_{2}$ ?

Let $U(2)$ denote the group of all invertible $2 \times 2$ complex matrices $A$ with $A \bar{A}^{\mathrm{T}}=I$, and let $S U(2)$ be the subgroup of $U(2)$ consisting of those matrices with determinant $1 .$

Determine the centre of $U(2)$.

Write down a surjective homomorphism from $U(2)$ to the group $T$ of all unit-length complex numbers whose kernel is $S U(2)$. Is $U(2)$ isomorphic to $S U(2) \times T$ ?

Paper 3, Section II, D

commentDefine the quotient group $G / H$, where $H$ is a normal subgroup of a group $G$. You should check that your definition is well-defined. Explain why, for $G$ finite, the greatest order of any element of $G / H$ is at most the greatest order of any element of $G$.

Show that a subgroup $H$ of a group $G$ is normal if and only if there is a homomorphism from $G$ to some group whose kernel is $H$.

A group is called metacyclic if it has a cyclic normal subgroup $H$ such that $G / H$ is cyclic. Show that every dihedral group is metacyclic.

Which groups of order 8 are metacyclic? Is $A_{4}$ metacyclic? For which $n \leqslant 5$ is $S_{n}$ metacyclic?

Paper 3, Section II, D

commentLet $g$ be an element of a group $G$. We define a map $g^{*}$ from $G$ to $G$ by sending $x$ to $g x g^{-1}$. Show that $g^{*}$ is an automorphism of $G$ (that is, an isomorphism from $G$ to $G$ ).

Now let $A$ denote the group of automorphisms of $G$ (with the group operation being composition), and define a map $\theta$ from $G$ to $A$ by setting $\theta(g)=g^{*}$. Show that $\theta$ is a homomorphism. What is the kernel of $\theta$ ?

Prove that the image of $\theta$ is a normal subgroup of $A$.

Show that if $G$ is cyclic then $A$ is abelian. If $G$ is abelian, must $A$ be abelian? Justify your answer.

Paper 3, Section II, D

commentDefine the sign of a permutation $\sigma \in S_{n}$. You should show that it is well-defined, and also that it is multiplicative (in other words, that it gives a homomorphism from $S_{n}$ to $\{\pm 1\})$.

Show also that (for $n \geqslant 2$ ) this is the only surjective homomorphism from $S_{n}$ to $\{\pm 1\}$.

Paper 3 , Section II, E

commentState Lagrange's theorem. Show that the order of an element $x$ in a finite group $G$ is finite and divides the order of $G$.

State Cauchy's theorem.

List all groups of order 8 up to isomorphism. Carefully justify that the groups on your list are pairwise non-isomorphic and that any group of order 8 is isomorphic to one on your list. [You may use without proof the Direct Product Theorem and the description of standard groups in terms of generators satisfying certain relations.]

Paper 3, Section I, E

commentWhat does it mean to say that $H$ is a normal subgroup of the group $G$ ? For a normal subgroup $H$ of $G$ define the quotient group $G / H$. [You do not need to verify that $G / H$ is a group.]

State the Isomorphism Theorem.

Let

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

be the group of $2 \times 2$ invertible upper-triangular real matrices. By considering a suitable homomorphism, show that the subset

$H=\left\{\left(\begin{array}{ll} 1 & b \\ 0 & 1 \end{array}\right) \mid b \in \mathbb{R}\right\}$

of $G$ is a normal subgroup of $G$ and identify the quotient $G / H$.

Paper 3, Section I, E

commentLet $w_{1}, w_{2}, w_{3}$ be distinct elements of $\mathbb{C} \cup\{\infty\}$. Write down the Möbius map $f$ that sends $w_{1}, w_{2}, w_{3}$ to $\infty, 0,1$, respectively. [Hint: You need to consider four cases.]

Now let $w_{4}$ be another element of $\mathbb{C} \cup\{\infty\}$ distinct from $w_{1}, w_{2}, w_{3}$. Define the cross-ratio $\left[w_{1}, w_{2}, w_{3}, w_{4}\right]$ in terms of $f$.

Prove that there is a circle or line through $w_{1}, w_{2}, w_{3}$ and $w_{4}$ if and only if the cross-ratio $\left[w_{1}, w_{2}, w_{3}, w_{4}\right]$ is real.

[You may assume without proof that Möbius maps map circles and lines to circles and lines and also that there is a unique circle or line through any three distinct points of $\mathbb{C} \cup\{\infty\} .]$

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

comment(a) Let $G$ be a finite group acting on a finite set $X$. State the Orbit-Stabiliser theorem. [Define the terms used.] Prove that

$\sum_{x \in X}|\operatorname{Stab}(x)|=n|G|$

where $n$ is the number of distinct orbits of $X$ under the action of $G$.

Let $S=\{(g, x) \in G \times X: g \cdot x=x\}$, and for $g \in G$, let $\operatorname{Fix}(g)=\{x \in X: g \cdot x=x\}$.

Show that

$|S|=\sum_{x \in X}|\operatorname{Stab}(x)|=\sum_{g \in G}|\operatorname{Fix}(g)|$

and deduce that

$n=\frac{1}{|G|} \sum_{g \in G}|\operatorname{Fix}(g)|$

(b) Let $H$ be the group of rotational symmetries of the cube. Show that $H$ has 24 elements. [If your proof involves calculating stabilisers, then you must carefully verify such calculations.]

Using $(*)$, find the number of distinct ways of colouring the faces of the cube red, green and blue, where two colourings are distinct if one cannot be obtained from the other by a rotation of the cube. [A colouring need not use all three colours.]

Paper 3, Section II, E

commentProve that every element of the symmetric group $S_{n}$ is a product of transpositions. [You may assume without proof that every permutation is the product of disjoint cycles.]

(a) Define the sign of a permutation in $S_{n}$, and prove that it is well defined. Define the alternating group $A_{n}$.

(b) Show that $S_{n}$ is generated by the set $\left\{\left(\begin{array}{lll}1 & 2\end{array}\right),\left(\begin{array}{llll}1 & 2 & 3 & \ldots\end{array}\right)\right\}$.

Given $1 \leqslant k<n$, prove that the set $\left\{\left(\begin{array}{ll}1 & \left.1+k),\left(\begin{array}{lll}1 & 2 & 3\end{array} \ldots n\right)\right\} \text { generates } S_{n} \text { if and only }\end{array}\right.\right.$ if $k$ and $n$ are coprime.

Paper 3, Section II, E

commentLet $N$ be a normal subgroup of a finite group $G$ of prime index $p=|G: N|$.

By considering a suitable homomorphism, show that if $H$ is a subgroup of $G$ that is not contained in $N$, then $H \cap N$ is a normal subgroup of $H$ of index $p$.

Let $C$ be a conjugacy class of $G$ that is contained in $N$. Prove that $C$ is either a conjugacy class in $N$ or is the disjoint union of $p$ conjugacy classes in $N$.

[You may use standard theorems without proof.]

Paper 3, Section I, D

commentState and prove Lagrange's theorem.

Let $p$ be an odd prime number, and let $G$ be a finite group of order $2 p$ which has a normal subgroup of order 2 . Show that $G$ is a cyclic group.

Paper 3, Section I, D

commentLet $G$ be a group, and let $H$ be a subgroup of $G$. Show that the following are equivalent.

(i) $a^{-1} b^{-1} a b \in H$ for all $a, b \in G$.

(ii) $H$ is a normal subgroup of $G$ and $G / H$ is abelian.

Hence find all abelian quotient groups of the dihedral group $D_{10}$ of order 10 .

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

commentState and prove the orbit-stabiliser theorem.

Let $p$ be a prime number, and $G$ be a finite group of order $p^{n}$ with $n \geqslant 1$. If $N$ is a non-trivial normal subgroup of $G$, show that $N \cap Z(G)$ contains a non-trivial element.

If $H$ is a proper subgroup of $G$, show that there is a $g \in G \backslash H$ such that $g^{-1} H g=H$.

[You may use Lagrange's theorem, provided you state it clearly.]

Paper 3, Section II, D

commentDefine the Möbius group $\mathcal{M}$ and its action on the Riemann sphere $\mathbb{C}_{\infty}$. [You are not required to verify the group axioms.] Show that there is a surjective group homomorphism $\phi: S L_{2}(\mathbb{C}) \rightarrow \mathcal{M}$, and find the kernel of $\phi .$

Show that if a non-trivial element of $\mathcal{M}$ has finite order, then it fixes precisely two points in $\mathbb{C}_{\infty}$. Hence show that any finite abelian subgroup of $\mathcal{M}$ is either cyclic or isomorphic to $C_{2} \times C_{2}$.

[You may use standard properties of the Möbius group, provided that you state them clearly.]

Paper 3, Section II, D

commentDefine the sign, $\operatorname{sgn}(\sigma)$, of a permutation $\sigma \in S_{n}$ and prove that it is well defined. Show that the function $\operatorname{sgn}: S_{n} \rightarrow\{1,-1\}$ is a homomorphism.

Show that there is an injective homomorphism $\psi: G L_{2}(\mathbb{Z} / 2 \mathbb{Z}) \rightarrow S_{4}$ such that $\operatorname{sgn} \circ \psi$ is non-trivial.

Show that there is an injective homomorphism $\phi: S_{n} \rightarrow G L_{n}(\mathbb{R})$ such that $\operatorname{det}(\phi(\sigma))=\operatorname{sgn}(\sigma) .$

Paper 3, Section II, D

commentFor each of the following, either give an example or show that none exists.

(i) A non-abelian group in which every non-trivial element has order $2 .$

(ii) A non-abelian group in which every non-trivial element has order 3 .

(iii) An element of $S_{9}$ of order 18 .

(iv) An element of $S_{9}$ of order 20 .

(v) A finite group which is not isomorphic to a subgroup of an alternating group.

Paper 3, Section I, D

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

Paper 3, Section I, D

commentSay 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.

Paper 3, Section II, D

comment(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.

Paper 3, Section II, D

comment(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 $G$.

Paper 3, Section II, D

commentLet $G, H$ 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 .

Paper 3, Section II, D

commentWhat 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 $U \in B_{r}$.

Paper 3, Section I, D

commentLet $G$ be a group, and suppose the centre of $G$ is trivial. If $p$ divides $|G|$, show that $G$ has a non-trivial conjugacy class whose order is prime to $p$.

Paper 3, Section I, D

commentLet $G=\mathbb{Q}$ be the rational numbers, with addition as the group operation. Let $x, y$ be non-zero elements of $G$, and let $N \leqslant G$ be the subgroup they generate. Show that $N$ is isomorphic to $\mathbb{Z}$.

Find non-zero elements $x, y \in \mathbb{R}$ which generate a subgroup that is not isomorphic to $\mathbb{Z}$.

Paper 3, Section II, D

comment(a) Let $G$ be a group, and $N$ a subgroup of $G$. Define what it means for $N$ to be normal in $G$, and show that if $N$ is normal then $G / N$ naturally has the structure of a group.

(b) For each of (i)-(iii) below, give an example of a non-trivial finite group $G$ and non-trivial normal subgroup $N \leqslant G$ satisfying the stated properties.

(i) $G / N \times N \simeq G$.

(ii) There is no group homomorphism $G / N \rightarrow G$ such that the composite $G / N \rightarrow G \rightarrow G / N$ is the identity.

(iii) There is a group homomorphism $i: G / N \rightarrow G$ such that the composite $G / N \rightarrow G \rightarrow G / N$ is the identity, but the map

$G / N \times N \rightarrow G, \quad(g N, n) \mapsto i(g N) n$

is not a group homomorphism.

Show also that for any $N \leqslant G$ satisfying (iii), this map is always a bijection.

Paper 3, Section II, D

commentLet $p$ be a prime number, and $G=G L_{2}\left(\mathbb{F}_{p}\right)$, the group of $2 \times 2$ invertible matrices with entries in the field $\mathbb{F}_{p}$ of integers modulo $p$.

The group $G$ acts on $X=\mathbb{F}_{p} \cup\{\infty\}$ by MÃ¶bius transformations,

$\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \cdot z=\frac{a z+b}{c z+d}$

(i) Show that given any distinct $x, y, z \in X$ there exists $g \in G$ such that $g \cdot 0=x$, $g \cdot 1=y$ and $g \cdot \infty=z$. How many such $g$ are there?

(ii) $G$ acts on $X \times X \times X$ by $g \cdot(x, y, z)=(g \cdot x, g \cdot y, g \cdot z)$. Describe the orbits, and for each orbit, determine its stabiliser, and the orders of the orbit and stabiliser.

Paper 3, Section II, D

commentLet $p$ be a prime number. Let $G$ be a group such that every non-identity element of $G$ has order $p$.

(i) Show that if $|G|$ is finite, then $|G|=p^{n}$ for some $n$. [You must prove any theorems that you use.]

(ii) Show that if $H \leqslant G$, and $x \notin H$, then $\langle x\rangle \cap H=\{1\}$.

Hence show that if $G$ is abelian, and $|G|$ is finite, then $G \simeq C_{p} \times \cdots \times C_{p}$.

(iii) 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 $a, b, x \in \mathbb{F}_{p}$ and $\mathbb{F}_{p}$ is the field of integers modulo $p$. Show that every nonidentity element of $G$ has order $p$ if and only if $p>2$. [You may assume that $G$ is a subgroup of the group of all $3 \times 3$ invertible matrices.]

Paper 3, Section II, D

commentLet $S_{n}$ be the group of permutations of $\{1, \ldots, n\}$, and suppose $n$ is even, $n \geqslant 4$.

Let $g=(12) \in S_{n}$, and $h=(12)(34) \ldots(n-1 n) \in S_{n}$.

(i) Compute the centraliser of $g$, and the orders of the centraliser of $g$ and of the centraliser of $h$.

(ii) Now let $n=6$. Let $G$ be the group of all symmetries of the cube, and $X$ the set of faces of the cube. Show that the action of $G$ on $X$ makes $G$ isomorphic to the centraliser of $h$ in $S_{6}$. [Hint: Show that $-1 \in G$ permutes the faces of the cube according to $h$.]

Show that $G$ is also isomorphic to the centraliser of $g$ in $S_{6}$.

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, 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, D

commentState and prove Lagrange's Theorem.

Show that the dihedral group of order $2 n$ has a subgroup of order $k$ for every $k$ dividing $2 n$.

Paper 3, Section I, D

comment(a) Let $G$ be the group of symmetries of the cube, and consider the action of $G$ on the set of edges of the cube. Determine the stabilizer of an edge and its orbit. Hence compute the order of $G$.

(b) The symmetric group $S_{n}$ acts on the set $X=\{1, \ldots, n\}$, and hence acts on $X \times X$ by $g(x, y)=(g x, g y)$. Determine the orbits of $S_{n}$ on $X \times X$.

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

comment(a) State the orbit-stabilizer theorem.

Let a group $G$ act on itself by conjugation. Define the centre $Z(G)$ of $G$, and show that $Z(G)$ consists of the orbits of size 1 . Show that $Z(G)$ is a normal subgroup of $G$.

(b) Now let $|G|=p^{n}$, where $p$ is a prime and $n \geqslant 1$. Show that if $G$ acts on a set $X$, and $Y$ is an orbit of this action, then either $|Y|=1$ or $p$ divides $|Y|$.

Show that $|Z(G)|>1$.

By considering the set of elements of $G$ that commute with a fixed element $x$ not in $Z(G)$, show that $Z(G)$ cannot have order $p^{n-1}$.

Paper 3, Section II, D

comment(a) Let $G$ be a finite group and let $H$ be a subgroup of $G$. Show that if $|G|=2|H|$ then $H$ is normal in $G$.

Show that the dihedral group $D_{2 n}$ of order $2 n$ has a normal subgroup different from both $D_{2 n}$ and $\{e\}$.

For each integer $k \geqslant 3$, give an example of a finite group $G$, and a subgroup $H$, such that $|G|=k|H|$ and $H$ is not normal in $G$.

(b) Show that $A_{5}$ is a simple group.

Paper 3, Section II, D

comment(a) Let

$S L_{2}(\mathbb{Z})=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \mid a d-b c=1, \quad a, b, c, d \in \mathbb{Z}\right\}$

and, for a prime $p$, let

$S L_{2}\left(\mathbb{F}_{p}\right)=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \mid a d-b c=1, \quad a, b, c, d \in \mathbb{F}_{p}\right\}$

where $\mathbb{F}_{p}$ consists of the elements $0,1, \ldots, p-1$, with addition and multiplication mod $p$.

Show that $S L_{2}(\mathbb{Z})$ and $S L_{2}\left(\mathbb{F}_{p}\right)$ are groups under matrix multiplication.

[You may assume that matrix multiplication is associative, and that the determinant of a product equals the product of the determinants.]

By defining a suitable homomorphism from $S L_{2}(\mathbb{Z}) \rightarrow S L_{2}\left(\mathbb{F}_{5}\right)$, show that

$\left\{\left(\begin{array}{cc} 1+5 a & 5 b \\ 5 c & 1+5 d \end{array}\right) \in S L_{2}(\mathbb{Z}) \mid a, b, c, d \in \mathbb{Z}\right\}$

is a normal subgroup of $S L_{2}(\mathbb{Z})$.

(b) Define the group $G L_{2}\left(\mathbb{F}_{5}\right)$, and show that it has order 480 . By defining a suitable homomorphism from $G L_{2}\left(\mathbb{F}_{5}\right)$ to another group, which should be specified, show that the order of $S L_{2}\left(\mathbb{F}_{5}\right)$ is 120 .

Find a subgroup of $G L_{2}\left(\mathbb{F}_{5}\right)$ of index 2 .

Paper 3, Section II, D

comment(a) Let $G$ be a finite group, and let $g \in G$. Define the order of $g$ and show it is finite. Show that if $g$ is conjugate to $h$, then $g$ and $h$ have the same order.

(b) Show that every $g \in S_{n}$ can be written as a product of disjoint cycles. For $g \in S_{n}$, describe the order of $g$ in terms of the cycle decomposition of $g$.

(c) Define the alternating group $A_{n}$. What is the condition on the cycle decomposition of $g \in S_{n}$ that characterises when $g \in A_{n}$ ?

(d) Show that, for every $n, A_{n+2}$ has a subgroup isomorphic to $S_{n}$.

Paper 3, Section I, D

commentExpress the element $(123)(234)$ in $S_{5}$ as a product of disjoint cycles. Show that it is in $A_{5}$. Write down the elements of its conjugacy class in $A_{5}$.

Paper 3, Section I, D

commentWrite down the matrix representing the following transformations of $\mathbb{R}^{3}$ :

(i) clockwise rotation of $45^{\circ}$ around the $x$ axis,

(ii) reflection in the plane $x=y$,

(iii) the result of first doing (i) and then (ii).

Paper 3, Section II, D

commentLet $G$ be a finite group, $X$ the set of proper subgroups of $G$. Show that conjugation defines an action of $G$ on $X$.

Let $B$ be a proper subgroup of $G$. Show that the orbit of $G$ on $X$ containing $B$ has size at most the index $|G: B|$. Show that there exists a $g \in G$ which is not conjugate to an element of $B$.

Paper 3, Section II, D

commentLet $G$ be a group, $X$ a set on which $G$ acts transitively, $B$ the stabilizer of a point $x \in X$.

Show that if $g \in G$ stabilizes the point $y \in X$, then there exists an $h \in G$ with $h g h^{-1} \in B$.

Let $G=S L_{2}(\mathbb{C})$, acting on $\mathbb{C} \cup\{\infty\}$ by MÃ¶bius transformations. Compute $B=G_{\infty}$, the stabilizer of $\infty$. Given

$g=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \in G$

compute the set of fixed points $\{x \in \mathbb{C} \cup\{\infty\} \mid g x=x\} .$

Show that every element of $G$ is conjugate to an element of $B$.

Paper 3, Section II, D

commentState Lagrange's theorem. Let $p$ be a prime number. Prove that every group of order $p$ is cyclic. Prove that every abelian group of order $p^{2}$ is isomorphic to either $C_{p} \times C_{p}$ or $C_{p^{2} \text {. }}$

Show that $D_{12}$, the dihedral group of order 12 , is not isomorphic to the alternating $\operatorname{group} A_{4}$.

Paper 3, Section II, D

comment(i) State the orbit-stabilizer theorem.

Let $G$ be the group of rotations of the cube, $X$ the set of faces. Identify the stabilizer of a face, and hence compute the order of $G$.

Describe the orbits of $G$ on the set $X \times X$ of pairs of faces.

(ii) Define what it means for a subgroup $N$ of $G$ to be normal. Show that $G$ has a normal subgroup of order 4 .

Paper 3 , Section II, D

commentLet $S(X)$ denote the group of permutations of a finite set $X$. Show that every permutation $\sigma \in S(X)$ can be written as a product of disjoint cycles. Explain briefly why two permutations in $S(X)$ are conjugate if and only if, when they are written as the product of disjoint cycles, they have the same number of cycles of length $n$ for each possible value of $n$.

Let $\ell(\sigma)$ denote the number of disjoint cycles, including 1-cycles, required when $\sigma$ is written as a product of disjoint cycles. Let $\tau$ be a transposition in $S(X)$ and $\sigma$ any permutation in $S(X)$. Prove that $\ell(\tau \sigma)=\ell(\sigma) \pm 1$.

Paper 3, Section I, D

commentState and prove Lagrange's theorem. Give an example to show that an integer $k$ may divide the order of a group $G$ without there being a subgroup of order $k$.

Paper 3, Section I, D

commentShow that every orthogonal $2 \times 2$ matrix $R$ is the product of at most two reflections in lines through the origin.

Every isometry of the Euclidean plane $\mathbb{R}^{2}$ can be written as the composition of an orthogonal matrix and a translation. Deduce from this that every isometry of the Euclidean plane $\mathbb{R}^{2}$ is a product of reflections.

Give an example of an isometry of $\mathbb{R}^{2}$ that is not the product of fewer than three reflections. Justify your answer.

Paper 3, Section II, D

commentWhat does it mean to say that a subgroup $K$ of a group $G$ is normal?

Let $\phi: G \rightarrow H$ be a group homomorphism. Is the kernel of $\phi$ always a subgroup of $G$ ? Is it always a normal subgroup? Is the image of $\phi$ always a subgroup of $H$ ? Is it always a normal subgroup? Justify your answers.

Let $\mathrm{SL}(2, \mathbb{Z})$ denote the set of $2 \times 2$ matrices $\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ with $a, b, c, d \in \mathbb{Z}$ and $a d-b c=1$. Show that $\mathrm{SL}(2, \mathbb{Z})$ is a group under matrix multiplication. Similarly, when $\mathbb{Z}_{2}$ denotes the integers modulo 2 , let $\mathrm{SL}\left(2, \mathbb{Z}_{2}\right)$ denote the set of $2 \times 2$ matrices $\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ with $a, b, c, d \in \mathbb{Z}_{2}$ and $a d-b c=1$. Show that $\mathrm{SL}\left(2, \mathbb{Z}_{2}\right)$ is also a group under matrix multiplication.

Let $f: \mathbb{Z} \rightarrow \mathbb{Z}_{2}$ send each integer to its residue modulo 2 . Show that

$\phi: \mathrm{SL}(2, \mathbb{Z}) \rightarrow \mathrm{SL}\left(2, \mathbb{Z}_{2}\right) ; \quad\left(\begin{array}{cc} a & b \\ c & d \end{array}\right) \mapsto\left(\begin{array}{ll} f(a) & f(b) \\ f(c) & f(d) \end{array}\right)$

is a group homomorphism. Show that the image of $\phi$ is isomorphic to a permutation group.

Paper 3, Section II, D

Define the cross-ratio $\left[a_{0}, a_{1}, a_{2}, z\right]$ of four points $a_{0}, a_{1}, a_{2}, z$ in $\mathbb{C} \cup\{\infty\}$, with $a_{0}, a_{1}, a_{2}$ distinct.

Let $a_{0}, a_{1}, a_{2}$ be three distinct points. Show that, for every value $w \in \mathbb{C} \cup\{\infty\}$, there is a unique point $z \in \mathbb{C} \cup\{\infty\}$ with $\left[a_{0}, a_{1}, a_{2}, z\right]=w$. Let $S$ be the set of points $z$ for which the cross-ratio $\left[a_{0}, a_{1}, a_{2}, z\right]$ is in $\mathbb{R} \cup\{\infty\}$. Show that $S$ is either a circle or else a straight line together with $\infty$.

A map $J: \mathbb{C} \cup\{\infty\} \rightarrow \mathbb{C} \cup\{\infty\}$ satisfies

$\left[a_{0}, a_{1}, a_{2}, J(z)\right]=\overline{\left[a_{0}, a_{1}, a_{2}, z\right]}$

for each value of $z$. Show that this gives a well-defined map $J$ with $J^{2}$ equal to the identity.

When the three points $a_{0}, a_{1}, a_{2}$ all lie on the real line, show that $J$ must be the conjugation map $J: z \mapsto \bar{z}$. Deduce from this that, for any three distinct points $a_{0}, a_{1}, a_{2}$