Chapter 1

Groups

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Problem 1.1

Let $x, y \in \mathbb {R}$ . Does the composition

x \bullet y := \sqrt[3]{x^{3}+y^{3}}

define a group?

Problem 1.2

Let $x, y \in \mathbb {R} \setminus \left\{ 0\right\}$ and $\cdot$ denotes multiplication in $\mathbb {R}$ . Does the composition

x \bullet y := \frac{x \cdot y}{2}

define a group?

Problem 1.3

Let $x, y \in \mathbb {R}$ . Is the composition $x \bullet y := \left|x+y\right|$ associative? Here $\left|\cdot \right|$ denotes the absolute value.

Problem 1.4

Consider the set

G = \left\{ (a,b) \in \mathbb {R}^{2} : a \neq 0\right\} .

We define the composition

(a,b) \bullet (c,d) := (ac, ad+b) .

Show that this composition defines a group. Is the group commutative?

Problem 1.5

Show that the set $\left\{ +1,-1,+i,-i\right\}$ forms a group under multiplication. Find all subgroups.

Problem 1.6

Let $i = \sqrt{-1}$ . Let $S$ be the set of complex numbers of the form $q+pi\sqrt{5}$ , where $p,q \in \mathbb {Q}$ and are not both simultaneously $0$ . Show that this set forms a group under multiplication of complex numbers.

Problem 1.7

Let $p$ be a prime number with $p \geq 3$ . Let $r$ and $s$ be rational numbers ( $r,s \in \mathbb {Q}$ ) with $r^{2}+s^{2}>0$ . Show that the set given by the numbers $r+s\sqrt{p}$ form a commutative group.

Problem 1.8

Show that the set

\left\{ e^{i\alpha } : \alpha \in \mathbb {R}\right\}

forms a group under multiplication. Note that $\left|e^{i\alpha }\right|=1$ .

Problem 1.9

Consider the additive group $(\mathbb {Z},+)$ . Give a proper subgroup.

Problem 1.10

Let $S := \mathbb {R} \setminus \left\{ -1\right\}$ . We define the binary operation on $S$

a \bullet b := a+b+ab .

Show that $\left\langle S, \bullet \right\rangle$ forms a group. Is the group commutative?

Problem 1.11

Let $G$ be a group and $x, y \in G$ . Show that $(xy)^{-1} = y^{-1}x^{-1}$ .

Problem 1.12

Let $S$ be the set of all rational numbers $\mathbb {Q}$ in the interval $0 \leq q < 1$ . Define the operation ( $q, p \in S$ )

q \bullet p := \begin{cases} q+p & \text{if } 0 \leq q+p < 1 \\ q+p-1 & \text{if } \quad q+p \geq 1 \end{cases} .

Show that $S$ with this operation is an abelian group.

Problem 1.13

Show that the finite set

\mathbb {Z}_{n} := \left\{ 0,1,\ldots ,n-1\right\}

for $n \geq 1$ forms an abelian group under addition modulo $n$ . The group is referred to as the group of integers modulo $n$ .

Problem 1.14

Let $n \in \mathbb {N}$ . Let $U(n)$ be the set of all positive integers less than $n$ and relatively prime to $n$ . Then $U(n)$ is a commutative group under multiplication modulo $n$ . Find the group table for $U(8)$ .

Problem 1.15

Consider the subset of odd integers

\left\{ 1,3,7,9,11,13,17,19\right\} .

Show that this set forms an abelian group under multiplication modulo $20$ .

Problem 1.16

Give the group table for the cyclic group $\mathbb {Z}_{6}$ of $6$ elements.

Problem 1.17

Consider the finite group $\mathbb {Z}_{2} \times \mathbb {Z}_{3}$ which has $2 \cdot 3 = 6$ elements given by $(0,0)$ , $(0,1)$ , $(0,2)$ , $(1,0)$ , $(1,1)$ , $(1,2)$ . The neutral element is $(0,0)$ . Show that $\mathbb {Z}_{2} \times \mathbb {Z}_{3}$ is cyclic.

Problem 1.18

Consider the functions defined on $\mathbb {R} \setminus \left\{ 0,1\right\}$

f_{1}(x)=x, \quad f_{2}(x)=\frac{1}{x}, \quad f_{3}(x)=1-x,

f_{4}(x)=\frac{x}{x-1}, \quad f_{5}(x)=\frac{1}{1-x}, \quad f_{6}(x)=1-\frac{1}{x} .

Show that these functions form a group with the function composition $f_{j} \circ f_{k}$ , where

(f_{j} \circ f_{k})(x) := f_{j}(f_{k}(x)) .

Problem 1.19

An isomorphism of a group $G$ with itself is an automorphism. Show that for each $g \in G$ the mapping $i_{g} : G \rightarrow G$ defined by

xi_{g} := g^{-1}xg

is an automorphism of $G$ , the inner automorphism of $G$ under conjugation by the group element $g$ . We have to show that $i_{g}$ is an isomorphism of $G$ with itself. Thus we have to show it is one to one, onto, and that

(xy)i_{g} = (xi_{g})(yi_{g})

for all $g \in G$ .

Problem 1.20

Let $C_{n}$ be the cyclic group. Show that $C_{6} \simeq C_{3} \times C_{2}$ .

Problem 1.21

(i)(ii)

(i)

Let $Z(G)$ be the center of the group $G$ . Show that $Z(G)$ is a commutative subgroup of $G$ .

(ii)

Let $a \in Z(G)$ and $g \in G$ . Then $ag=ga$ . Show that $a^{-1}g = ga^{-1}$ .

Problem 1.22

Show that the center $Z(G)$ of a group $G$ is a normal subgroup of the group $G$ .

Problem 1.23

(i)(ii)

Let $H$ be a subgroup of a group $G$ . The normalizer $N(H)$ of $H$ in $G$ is defined by

N(H) := \left\{ g \in G : gHg^{-1} = H\right\} .

(i)

Show that $N(H)$ is a subgroup of $G$ .

(ii)

Show that $H$ is a normal subgroup of $N(H)$ .

Problem 1.24

Consider a group $G$ . We define the group commutator ( $x,y \in G$ )

[x,y] := xyx^{-1}y^{-1}

Consider the two $2 \times 2$ matrices with determinant $1$

x = \begin{pmatrix} t & 1 \\ -1 & 0 \end{pmatrix}, \qquad y = \begin{pmatrix} 1 & b \\ c & 1+bc \end{pmatrix}

where $t, b, c \in \mathbb {R}$ . This means $x$ and $y$ are elements of $SL(2,\mathbb {R})$ . Find the group commutator.

Problem 1.25

Let $G$ be a group. Given two elements $g_{1}, g_{2} \in G$ . One defines the group commutator of $g_{1}$ and $g_{2}$ to be the element

g_{1}g_{2}g_{1}^{-1}g_{2}^{-1} .

Consider the compact Lie group $SO(2,\mathbb {R})$ with

g_{1}(\alpha ) = \begin{pmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{pmatrix}, \qquad g_{1}(\beta ) = \begin{pmatrix} \cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{pmatrix}

Find the group commutator.

Problem 1.26

(i)(ii)

(i)

Show that the set of all group commutators $aba^{-1}b^{-1}$ of a group $G$ generates a normal subgroup $G'$ (the so-called commutator subgroup) of $G$ .

(ii)

Show that $G/G'$ is abelian.

Problem 1.27

Let $H$ and $K$ be two subgroups of a group $G$ . Show that the relation

g \sim g' \quad \text{if} \quad g' = hgk \quad \text{for some} \quad h \in H, \ k \in K

is an equivalence relation, partitioning $G$ into double cosets $Hg_{j}K$ for $g_{j} \in G$ .

Problem 1.28

Let $G$ be a group with the composition $\bullet$ . Let $H$ be a subgroup of $G$ . We define the relation

a \sim b \Leftrightarrow a \bullet b^{-1} \in H \quad \text{for} \quad a,b \in G .

Is this relation an equivalence relation?

Problem 1.29

Let $G$ be a group. Let $G_{1}$ and $G_{2}$ be two subgroups of $G$ . Show that the intersection of $G_{1}$ and $G_{2}$ is itself a subgroup.

Problem 1.30

The set $\left\{ 1,-1\right\}$ forms a group $G_{1}$ under multiplication. The set $\left\{ 0,1\right\}$ forms a group $G_{2}$ under the XOR-operation $\oplus$ , i.e.

0 \oplus 0 = 0, \quad 0 \oplus 1 = 1, \quad 1 \oplus 0 = 1, \quad 1 \oplus 1 = 0 .

Show that the two groups are isomorphic.

Problem 1.31

Show that any infinite cyclic group $G$ is isomorphic to the group $\mathbb {Z}$ of integers under addition.

Problem 1.32

(i)(ii)

Let $\phi_{1} : G_{1} \rightarrow G_{2}$ and $\phi_{2} : G_{2} \rightarrow G_{1}$ be homomorphisms such that $\phi_{1}\phi_{2} = \phi_{2}\phi_{1} = i$ , where $i$ is the identity map. This means $\phi_{1}\phi_{2} : G_{1} \rightarrow G_{1}$ and $\phi_{2}\phi_{1} : G_{2} \rightarrow G_{2}$ are both the identity map. The First Isomorphism Theorem states that the image of a group homomorphism, $\mathrm{im}(\phi_{1})$ is isomorphic to the quotient group $G_{1}/\ker (\phi_{1})$ .

(i)

Show that both $\phi_{1}$ and $\phi_{2}$ are isomorphisms of $G_{1}$ with $G_{2}$ .

(ii)

Show that $\phi_{1} = (\phi_{2})^{-1}$ .

Problem 1.33

Show that the finite groups $G_{1} := \left\{ +1,-1 : \cdot \right\}$ and

G_{2} = \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} : *\right\}

are isomorphic, where $*$ denotes matrix multiplication.

Problem 1.34

(i)(ii)

Consider the matrix groups with matrix multiplication

G = \left\{ \begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} : a,b \in \mathbb {R}, \ a>0\right\} , \qquad N = \left\{ \begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} : b \in \mathbb {R}\right\} .

(i)

Show that $N$ is a normal subgroup of $G$ .

(ii)

Show that the factor group $G/N$ is isomorphic to the additive group $(\mathbb {R},+)$ .

Problem 1.35

Let $G$ be a group. Let $H$ be a subgroup of $G$ . For any fixed $g \in G$ one defines

gHg^{-1} := \left\{ ghg^{-1} : h \in H\right\} .

The group $gHg^{-1}$ is called a conjugate of the subgroup $H$ . Show that if $H$ is cyclic, then $gHg^{-1}$ is cyclic.

Problem 1.36

Let $G_{1}, G_{2}, \ldots , G_{n}$ be groups. For

(g_{1},g_{2},\ldots ,g_{n}) \in G_{1} \times G_{2} \times \cdots \times G_{n}, \qquad (h_{1},h_{2},\ldots ,h_{n}) \in G_{1} \times G_{2} \times \cdots \times G_{n}

we define $(g_{1},g_{2},\ldots ,g_{n}) \bullet (h_{1},h_{2},\ldots ,h_{n})$ to be $(g_{1}h_{1}, g_{2}h_{2}, \ldots , g_{n}h_{n})$ . Show that $G_{1} \times G_{2} \times \cdots \times G_{n}$ is a group under this composition (called the external direct product of the groups $G_{j}$ ).

Problem 1.37

Let $G$ be a finite group. A conjugacy class for a given element $g$ of $G$ is the set defined by

C(g) := \left\{ aga^{-1} : a \in G\right\} .

Show that the number of elements in a conjugacy class divides the order of $G$ .

Problem 1.38

The $3 \times 3$ permutation matrices

\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}

form a group under matrix multiplication. Find the conjugacy classes.

Problem 1.39

Let $D_{n}$ be the dihedral group with $n \geq 3$ . This is the finite nonabelian group of rigid motions of a regular $n$ -gon. The order of the group is $2n$ . Find the center

Z(D_{n}) := \left\{ c \in D_{n} : cx=xc \quad \text{for all} \quad x \in D_{n}\right\} .

Problem 1.40

The number of elements of a group (finite or infinite) is called its order. The order of $G$ is denoted by $\left|G\right|$ . The order of an element $g$ in the group $G$ is the smallest positive integer $k$ such that $g^{k}=e$ , where $e$ is the identity element of the group. It is denoted by $\left|g\right|$ . If no such element exists, one says that $g$ has infinite order. Consider the $3 \times 3$ permutation matrix

P = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}

and matrix multiplication. What is the order of $P$ in the group of permutation matrices? What is order of the group of all $3 \times 3$ permutation matrices?

Problem 1.41

Consider the set of Pauli spin matrices $\sigma_{x}, \sigma_{y}, \sigma_{z}$

\sigma _{x} := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma _{y} := \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \sigma _{z} := \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

and the $2 \times 2$ identity matrix $I_{2}$ . Can we extend this set so that we obtain a group under matrix multiplication?

Problem 1.42

The group $SL(2,\mathbb {R})$ consists of all $2 \times 2$ matrices over $\mathbb {R}$ with determinant equal to $1$ . Let

A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix}

be elements of $SL(2,\mathbb {R})$ . What is the order of $A$ , $B$ and $AB$ ?

Problem 1.43

(i)(ii)(iii)

Consider the set $N = \left\{ 1,2,\ldots ,n\right\}$ . The set of all permutations of $N$ is called the symmetric group of degree $n$ and is denoted by $S_{n}$ . The elements of $S_{n}$ have the form

s = \begin{pmatrix} 1 & 2 & \ldots & n \\ s(1) & s(2) & \ldots & s(n) \end{pmatrix} .

The order of the group is $n!$ .

(i)

Consider $S_{3}$ . Give all the elements of this group.

(ii)

Is the group commutative?

(iii)

How many subgroups does $S_{3}$ have?

Problem 1.44

Consider the square where the corners are numbered counter-clockwise $1, 2, 3, 4$ . Show that the symmetry group $D_{4}$ of the square is a subgroup of the symmetric group $S_{4}$ . The group $S_{4}$ is defined as the set of all permutations of $\left\{ 1,2,3,4\right\}$ . Each element of the symmetry group $D_{4}$ transforms every corner of the square to another corner. Thus it transforms every element of $\left\{ 1,2,3,4\right\}$ (the number of the corner) to another element.

Problem 1.45

The group of all even permutations of $n$ objects is known as the alternating group of degree $n$ and is denoted by $A_{n}$ . For $n>1$ , the alternating group has order $n!/2$ . Find all elements of the group $A_{3}$ .

Problem 1.46

If $G$ is a group and $g \in G$ , then $H := \left\{ g^{n} : n \in \mathbb {Z}\right\}$ is a subgroup of $G$ . This group is the cyclic subgroup of $G$ generated by $g$ . Consider the $3 \times 3$ permutation matrix

A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} .

Show that $A, A^{2}, A^{3}$ form a group under matrix multiplication.

Problem 1.47

Consider the $4 \times 4$ permutation matrix

A = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix} .

Let $I_{4}$ be the $4 \times 4$ identity matrix. Find $A^{2}, A^{3}, A^{4}$ and thus show that we have a cyclic group.

Problem 1.48

(i)(ii)(iii)

(i)

Consider the $4 \times 4$ matrix

A = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix} .

Obviously $\operatorname {rk}\left(A\right)=4$ and therefore the inverse exists. Let $I_{4}$ be the $4 \times 4$ identity matrix. Find the inverse of $A$ .

(ii)

Does the set $\left\{ A, A^{-1}, I_{4}\right\}$ form a group under matrix multiplication? If not can we find a finite extension to the set to obtain a group?

(iii)

Calculate the determinants of the matrices found in (ii). Show that these numbers form a group under multiplication.

Problem 1.49

Show that the following matrices form a group under matrix multiplication

s_{1} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad s_{2} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \quad s_{3} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix},

s_{4} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, \quad s_{5} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \quad s_{6} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} .

These matrices are the six $3 \times 3$ permutation matrices.

Problem 1.50

(i)(ii)

Given two manifolds $M$ and $N$ , a bijective map $\phi$ from $M$ to $N$ is called a diffeomorphism if both $\phi : M \rightarrow N$ and its inverse $\phi^{-1}$ are differentiable. Let $f : \mathbb {R} \rightarrow \mathbb {R}$ be given by the analytic function

f(x) = 4x(1-x)

and the analytic function $g : \mathbb {R} \rightarrow \mathbb {R}$ be given by

g(x) = 1-2x^{2} .

(i)

Can one find a diffeomorphism $\phi : \mathbb {R} \rightarrow \mathbb {R}$ such that

g = \phi \circ f \circ \phi ^{-1} \ ?

(ii)

Consider the diffeomorphism $\psi : \mathbb {R} \rightarrow \mathbb {R}$

\psi (x) = \sinh (x) .

Calculate $\psi \circ g \circ \psi^{-1}$ .

Problem 1.51

Let $G$ be the dihedral group defined by

D_{8} := \left\langle a,b : a^{4}=b^{2}=e, \ b^{-1}ab=a^{-1}\right\rangle

where $e$ is the identity element in the group. Define the invertible $2 \times 2$ matrices

A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} .

Let $I_{2}$ be the $2 \times 2$ identity matrix. Show that $A^{4} = B^{2} = I_{2}$ , $B^{-1}AB = A^{-1}$ and thus show that we have a representation of the dihedral group.

Problem 1.52

Consider the $2 \times 2$ matrices

C_{3} = \begin{pmatrix} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2 & -1/2 \end{pmatrix}, \qquad \sigma _{3} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

together with the $2 \times 2$ identity matrix $I_{2}$ . Do these matrices form a group under matrix multiplication? If not find the smallest set of matrices with $C_{3}$ , $\sigma_{3}$ and $I_{2}$ as elements which forms a group under matrix multiplication.

Problem 1.53

Let $c \in \mathbb {R}$ and $c \neq 0$ . Show that the $2 \times 2$ matrices

A(c) = \begin{pmatrix} c & c \\ c & c \end{pmatrix}

form a group under matrix multiplication. Notice that the matrices $A(c)$ are not invertible in the usual sense, i.e. $\operatorname {det}\left(A(c)\right)=0$ . Find $\mathrm{exp}\left(A(c)\right)$ .

Problem 1.54

(i)(ii)

(i)

Find all $2 \times 2$ matrices $A$ over $\mathbb {R}$ with

\operatorname {det}\left(A\right) = a_{11}a_{22}-a_{12}a_{21} = 1

and

A\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} .

(ii)

Do these matrices form a group under matrix multiplication?

Problem 1.55

Let $G$ be a group. If a subset $H$ of $G$ itself is a group under the operation of $G$ , then $H$ is called a subgroup of $G$ . Consider the group of the $3 \times 3$ permutation matrices and matrix multiplication. Find a subgroup of order $3$ .

Problem 1.56

Consider finite groups. Then Lagrange’s theorem tells us that the order of a subgroup of a finite group is always a divisor of the order of the group. Show that we cannot conclude that if $G$ is a finite group and if $m$ is a divisor of the order of $G$ that $G$ will contain a subgroup of order $m$ . Consider the subgroup $A_{4}$ with determinant $+1$ of the group of all $4 \times 4$ permutation matrices. The order of this group is $4!/2=12$ .

Problem 1.57

Do the twelve $2 \times 2$ orthogonal matrices

\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \quad \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix},

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix},

\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \quad \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}, \quad \frac{1}{\sqrt{2}}\begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix}, \quad \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}

form a group under matrix multiplication? If not add the matrices so that one has a group.

Problem 1.58

Consider the group $GL(n,\mathbb {R})$ of all invertible $n \times n$ matrices over $\mathbb {R}$ . Show that the $n \times n$ matrices with integer entries and determinant $+1$ or $-1$ form a subgroup of $GL(n,\mathbb {R})$ .

Problem 1.59

(i)(ii)

Consider the group $G = \left\{ a,b,c\right\}$ with group operation $\cdot : G \times G \rightarrow G$ defined by the table

\begin{array}{c|ccc} \cdot & a & b & c \\ \hline a & a & b & c \\ b & b & c & a \\ c & c & a & b \end{array} .

Let $M(n)$ denote the vector space of $n \times n$ matrices over $\mathbb {C}$ .

(i)

Show that $f : G \rightarrow M(3)$ is a faithful representation for $(G,\cdot )$ where

f(a) := \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

f(b) := \begin{pmatrix} \frac{\sqrt{3}}{2}i-\frac{1}{2} & 0 & 0 \\ 0 & -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ 0 & -\frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix}

f(c) := \begin{pmatrix} -\frac{\sqrt{3}}{2}i-\frac{1}{2} & 0 & 0 \\ 0 & -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix} .

(ii)

Is the representation in (i) reducible? Prove or disprove.

Problem 1.60

The direct sum of the $2 \times 2$ matrices $A, B$ is defined as

A \oplus B := \begin{pmatrix} A & 0_{2} \\ 0_{2} & B \end{pmatrix}

where $0_{2}$ is the $2 \times 2$ zero matrix. The $2 \times 2$ matrices

I_{2} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad N = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

form a group under matrix multiplication. Show that the $4 \times 4$ matrices

I_{2} \oplus I_{2}, \quad I_{2} \oplus N, \quad N \oplus I_{2}, \quad N \oplus N

form a group under matrix multiplication.

Problem 1.61

Let $A, B$ be $2 \times 2$ matrices. We define the composition

A \star B := \begin{pmatrix} a_{11} & 0 & 0 & a_{12} \\ 0 & b_{11} & b_{12} & 0 \\ 0 & b_{21} & b_{22} & 0 \\ a_{21} & 0 & 0 & a_{22} \end{pmatrix} .

Let

E = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad N = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} .

Show that the $4 \times 4$ matrices $E \star E$ , $E \star N$ , $N \star E$ , $N \star N$ form a group under matrix multiplication.

Problem 1.62

The Kronecker product of the $2 \times 2$ matrices $A, B$ is defined as the $4 \times 4$ matrix

A \otimes B := \begin{pmatrix} a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \end{pmatrix} .

Let

N = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} .

Show that the $4 \times 4$ matrices $I_{2} \otimes I_{2}$ , $I_{2} \otimes N$ , $N \otimes I_{2}$ , $N \otimes N$ form a group under matrix multiplication.

Problem 1.63

Which of the $4 \times 4$ permutation matrices can be written as Kronecker products of $2 \times 2$ permutation matrices?

Problem 1.64

Show that the four $2 \times 2$ matrices

A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}, \quad C = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad D = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}

form a group under matrix multiplication. Is the group abelian?

Problem 1.65

Consider the set of all $2 \times 2$ matrices over the set of integers $\mathbb {Z}$ with determinant equal to $1$ . Show that these matrices form a group under matrix multiplication. This group is called $SL(2,\mathbb {Z})$ .

Problem 1.66

Consider the group of all $4 \times 4$ permutation matrices

A = \begin{pmatrix} 1& 0& 0& 0 \\ 0& 1& 0& 0 \\ 0& 0& 1& 0 \\ 0& 0& 0& 1 \end{pmatrix}, \ B = \begin{pmatrix} 1& 0& 0& 0 \\ 0& 1& 0& 0 \\ 0& 0& 0& 1 \\ 0& 0& 1& 0 \end{pmatrix}, \ C = \begin{pmatrix} 1& 0& 0& 0 \\ 0& 0& 1& 0 \\ 0& 1& 0& 0 \\ 0& 0& 0& 1 \end{pmatrix},

D = \begin{pmatrix} 1& 0& 0& 0 \\ 0& 0& 1& 0 \\ 0& 0& 0& 1 \\ 0& 1& 0& 0 \end{pmatrix}, \ E = \begin{pmatrix} 1& 0& 0& 0 \\ 0& 0& 0& 1 \\ 0& 1& 0& 0 \\ 0& 0& 1& 0 \end{pmatrix}, \ F = \begin{pmatrix} 1& 0& 0& 0 \\ 0& 0& 0& 1 \\ 0& 0& 1& 0 \\ 0& 1& 0& 0 \end{pmatrix},

G = \begin{pmatrix} 0& 1& 0& 0 \\ 1& 0& 0& 0 \\ 0& 0& 1& 0 \\ 0& 0& 0& 1 \end{pmatrix}, \ H = \begin{pmatrix} 0& 1& 0& 0 \\ 1& 0& 0& 0 \\ 0& 0& 0& 1 \\ 0& 0& 1& 0 \end{pmatrix}, \ I = \begin{pmatrix} 0& 1& 0& 0 \\ 0& 0& 1& 0 \\ 1& 0& 0& 0 \\ 0& 0& 0& 1 \end{pmatrix},

J = \begin{pmatrix} 0& 1& 0& 0 \\ 0& 0& 1& 0 \\ 0& 0& 0& 1 \\ 1& 0& 0& 0 \end{pmatrix}, \ K = \begin{pmatrix} 0& 1& 0& 0 \\ 0& 0& 0& 1 \\ 1& 0& 0& 0 \\ 0& 0& 1& 0 \end{pmatrix}, \ L = \begin{pmatrix} 0& 1& 0& 0 \\ 0& 0& 0& 1 \\ 0& 0& 1& 0 \\ 1& 0& 0& 0 \end{pmatrix},

M = \begin{pmatrix} 0& 0& 1& 0 \\ 1& 0& 0& 0 \\ 0& 1& 0& 0 \\ 0& 0& 0& 1 \end{pmatrix}, \ N = \begin{pmatrix} 0& 0& 1& 0 \\ 1& 0& 0& 0 \\ 0& 0& 0& 1 \\ 0& 1& 0& 0 \end{pmatrix}, \ O = \begin{pmatrix} 0& 0& 1& 0 \\ 0& 1& 0& 0 \\ 1& 0& 0& 0 \\ 0& 0& 0& 1 \end{pmatrix},

P = \begin{pmatrix} 0& 0& 1& 0 \\ 0& 1& 0& 0 \\ 0& 0& 0& 1 \\ 1& 0& 0& 0 \end{pmatrix}, \ Q = \begin{pmatrix} 0& 0& 1& 0 \\ 0& 0& 0& 1 \\ 1& 0& 0& 0 \\ 0& 1& 0& 0 \end{pmatrix}, \ R = \begin{pmatrix} 0& 0& 1& 0 \\ 0& 0& 0& 1 \\ 0& 1& 0& 0 \\ 1& 0& 0& 0 \end{pmatrix},

S = \begin{pmatrix} 0& 0& 0& 1 \\ 1& 0& 0& 0 \\ 0& 1& 0& 0 \\ 0& 0& 1& 0 \end{pmatrix}, \ T = \begin{pmatrix} 0& 0& 0& 1 \\ 1& 0& 0& 0 \\ 0& 0& 1& 0 \\ 0& 1& 0& 0 \end{pmatrix}, \ U = \begin{pmatrix} 0& 0& 0& 1 \\ 0& 1& 0& 0 \\ 1& 0& 0& 0 \\ 0& 0& 1& 0 \end{pmatrix},

V = \begin{pmatrix} 0& 0& 0& 1 \\ 0& 1& 0& 0 \\ 0& 0& 1& 0 \\ 1& 0& 0& 0 \end{pmatrix}, \ W = \begin{pmatrix} 0& 0& 0& 1 \\ 0& 0& 1& 0 \\ 1& 0& 0& 0 \\ 0& 1& 0& 0 \end{pmatrix}, \ X = \begin{pmatrix} 0& 0& 0& 1 \\ 0& 0& 1& 0 \\ 0& 1& 0& 0 \\ 1& 0& 0& 0 \end{pmatrix} .

Find all subgroups and discuss whether they are commutative or not.

Problem 1.67

(i)(ii)

The Heisenberg group is the group of $3 \times 3$ upper triangular matrices of the form

\begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \end{pmatrix}

where $a_{jk} \in \mathbb {F}$ with $\mathbb {F}$ an arbitrary field. In the following we consider $\mathbb {F} = \mathbb {C}$ .

(i)

Is the Heisenberg group commutative?

(ii)

Let $A, B$ be elements of the Heisenberg group. Does the braid like relation $ABBA = BAAB$ hold?

Problem 1.68

The group $SL(2,\mathbb {Z})$ is generated by

T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \qquad S = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} .

Find $A \in SL(2,\mathbb {Z})$ in terms of $T$ and $S$ with

A = \begin{pmatrix} -2 & 1 \\ -1 & 0 \end{pmatrix} .

Problem 1.69

Consider the symmetric group $S_{3}$ of the $3 \times 3$ permutation matrices. Do the group elements which satisfy the condition $P_{j}^{2} = I_{3}$ form a subgroup of $S_{3}$ ?

Problem 1.70

(i)(ii)

(i)

Consider the group $G$ of all $3 \times 3$ permutation matrices. Show that

\Pi = \frac{1}{\left|G\right|} \sum _{g \in G} g

is a projection matrix, i.e. $\Pi = \Pi^{*}$ and $\Pi^{2} = \Pi$ . Here $\left|G\right|$ denotes the number of elements in the group.

(ii)

Consider the subgroup $H$ of $G$ given by the matrices

\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} .

Show that

\Pi = \frac{1}{\left|H\right|} \sum _{g \in H} g

is a projection matrix.

Problem 1.71

Consider the permutation group $S_{n}$ . Let $\sigma \in S_{n}$ . Is $\varphi (\sigma ) = \operatorname {sgn}(\sigma )$ a homomorphism? Here $\operatorname {sgn}(\sigma )$ denotes the sign of a permutation $\sigma$ .

Problem 1.72

(i)(ii)

(i)

Consider the permutation group $S_{n}$ . The alternating group $A_{n}$ consists of all the even permutations of $S_{n}$ . Show that they form a group under composition.

(ii)

Consider $S_{3}$ with the matrix representation by the $3 \times 3$ permutation matrices

\begin{pmatrix} 1& 0& 0 \\ 0& 1& 0 \\ 0& 0& 1 \end{pmatrix}, \quad \begin{pmatrix} 1& 0& 0 \\ 0& 0& 1 \\ 0& 1& 0 \end{pmatrix}, \quad \begin{pmatrix} 0& 1& 0 \\ 1& 0& 0 \\ 0& 0& 1 \end{pmatrix},

\begin{pmatrix} 0& 1& 0 \\ 0& 0& 1 \\ 1& 0& 0 \end{pmatrix}, \quad \begin{pmatrix} 0& 0& 1 \\ 1& 0& 0 \\ 0& 1& 0 \end{pmatrix}, \quad \begin{pmatrix} 0& 0& 1 \\ 0& 1& 0 \\ 1& 0& 0 \end{pmatrix} .

Find the matrix representation of $A_{3}$ using the $3 \times 3$ permutation matrices.

Problem 1.73

Let $x \in \mathbb {R}$ and $z \in \mathbb {C}$ . Show that the triangular $3 \times 3$ matrices

g(z,x) = \begin{pmatrix} 1 & \bar{z} & \frac{1}{2}\bar{z}z+ix \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix}

form a group under matrix multiplication.

Problem 1.74

Let $x \in \mathbb {R}$ and $z_{1}, z_{2} \in \mathbb {C}$ . We set $\overrightarrow {z} = (z_{1},z_{2})^{T}$ . Show that the triangular $4 \times 4$ matrices

g(\overrightarrow {z},x) = \begin{pmatrix} 1 & \bar{z}_{1} & \bar{z}_{2} & \frac{1}{2}(\overrightarrow {z}|\overrightarrow {z})+ix \\ 0 & 1 & 0 & z_{1} \\ 0 & 0 & 1 & z_{2} \\ 0 & 0 & 0 & 1 \end{pmatrix}

form a group under matrix multiplication, where $(\overrightarrow {z}|\overrightarrow {z})$ is the scalar product

(\overrightarrow {z}|\overrightarrow {z}) \equiv \bar{\overrightarrow {z}}\overrightarrow {z} := \left|z_{1}\right|^{2}+\left|z_{2}\right|^{2} .

Problem 1.75

All invertible $n \times n$ matrices form a group under matrix multiplication. Let $a, b, c \in \mathbb {R}$ . Consider the $4 \times 4$ matrix

M = \frac{i}{4} \begin{pmatrix} -2c & -a+ib & -a+ib & 0 \\ -2a-2ib & c-a & c+a & 2b-2ic \\ -2a-2ib & c+a & c-a & -2b+2ic \\ 0 & b+ic & -b-ic & 2a \end{pmatrix} .

Let $T$ be the invertible $4 \times 4$ matrix

T = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1/2 & 1/2 & 0 \\ 0 & -1/2 & 1/2 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} .

Show that the $4 \times 4$ matrix $TMT^{-1}$ can be written as a direct sum of two $2 \times 2$ matrices.

Problem 1.76

(i)(ii)

(i)

Show that the cyclic group $C_{3}$ is generated by

A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}

under matrix multiplication. Are the matrices linearly independent?

(ii)

Let $r_{0}, r_{1}, r_{2}, r_{3} \in \mathbb {R}$ . Consider the set of all linear combinations of the form

r_{0}I_{2}+r_{1}A+r_{2}A^{2}+r_{3}A^{3} .

Show that this set is a ring.

Problem 1.77

(i)(ii)(iii)

Let $L$ be the invertible matrix

L = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} .

(i)

Find all invertible $2 \times 2$ matrices $M$ such that $M^{-1}LM = L$ .

(ii)

Find all invertible $2 \times 2$ matrices $S$ such that $S^{-1}LS = L^{-1}$ .

(iii)

Calculate the matrix $(MS)^{-1}L(MS)$ . Discuss.

Problem 1.78

Consider the square. Find all symmetry operations which leave the square invariant. Use permutation matrices.

Problem 1.79

(i)(ii)

Consider the group $GL(2,\mathbb {F})$ , where the underlying field $\mathbb {F}$ could be $\mathbb {Q}$ , $\mathbb {R}$ , $\mathbb {C}$ or $\mathbb {Z}_{p}$ , where $p$ is prime. Consider the $2 \times 2$ matrix

A = \begin{pmatrix} 3 & 2 \\ 1 & 5 \end{pmatrix} .

(i)

Find the inverse of $A$ if the underlying field is $\mathbb {Q}$ .

(ii)

Find the inverse of $A$ if the underlying field is $\mathbb {Z}_{5}$ .

Problem 1.80

Show that $SL(2,\mathbb {R})$ is a normal subgroup of $GL(2,\mathbb {R})$ . The group $SL(2,\mathbb {R})$ consists of all $2 \times 2$ matrices over $\mathbb {R}$ with determinant $1$ . Use the following theorem.

Remark.

A subgroup $H$ of $G$ is normal in $G$ if and only if $gHg^{-1} \subseteq H$ for all $g \in G$ .

Problem 1.81

Consider the group $G = GL(2,\mathbb {R})$ and the normal subgroup $H = SL(2,\mathbb {R})$ . Let $g \in G$ with $\operatorname {det}\left(g\right)=2$ . Show that the set $gH$ is the set of all $2 \times 2$ matrices in $G$ with determinant $2$ .

Problem 1.82

Let $G$ be a group and let $g \in G$ . Show that

H := \left\{ g^{n} : n \in \mathbb {Z}\right\}

is a subgroup of $G$ and is the smallest subgroup of $G$ that contains the element $g$ .

Problem 1.83

The Klein four-group is an abelian group of order $4$ . It is the smallest non-cyclic group. The Klein four-group consists of the identity element $e$ and the elements $i, j, k$ with the composition

i^{2} = j^{2} = k^{2} = ijk = e .

Give a matrix representations with $2 \times 2$ matrices.

Problem 1.84

Let $\omega = \exp (i2\pi /3)$ , $\bar{\omega } = \exp (-i2\pi /3)$ . Show that the unitary matrix

U = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & \omega & \bar{\omega } \\ 1 & \bar{\omega } & \omega \\ 1 & 1 & 1 \end{pmatrix}

reduces the natural representation of $C_{3}$

\begin{pmatrix} 1& 0& 0 \\ 0& 1& 0 \\ 0& 0& 1 \end{pmatrix}, \quad \begin{pmatrix} 0& 1& 0 \\ 0& 0& 1 \\ 1& 0& 0 \end{pmatrix}, \quad \begin{pmatrix} 0& 0& 1 \\ 1& 0& 0 \\ 0& 1& 0 \end{pmatrix}

to its irreducible form.

Problem 1.85

Let $a, b \in \mathbb {R}$ . What are the conditions on $a, b$ such that the matrices

\begin{pmatrix} a & b & 0 \\ b & a & b \\ 0 & b & a \end{pmatrix}

form a group under matrix multiplication?

Problem 1.86

Let $a, b, c \in \mathbb {R}$ . Show that the matrices

\begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}

form a non-commutative group under matrix multiplication. Find the center of the group $G$ .

Problem 1.87

Let $A_{n}$ be the alternating group. The order of $A_{n}$ is $n!/2$ . Is $A_{n}$ a normal subgroup of $S_{n}$ ?

Problem 1.88

Let $S$ be a set and $G$ be a group. An action of $G$ on $S$ is a map

\varphi : G \times S \rightarrow S, \qquad \varphi (g,x) = gx

such that

$ex = x$ for the identity element $e$ of $G$ and for all $x \in S$
$(g_{1}g_{2})x = g_{1}(g_{2}x)$ for all $x \in S$ and all $g_{1}, g_{2} \in G$ . Under these conditions, $S$ is a $G$ -set. Give an example for $G$ , $S$ and the action.

Problem 1.89

(i)(ii)(iii)

(i)

Show that the set of matrices

G_{0} = \left\{ \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} : a \neq 0\right\} \subset GL(2,\mathbb {C})

form a group under matrix multiplication.

(ii)

Is the group commutative?

(iii)

Find the center of the group.

Problem 1.90

The group $SL(2,\mathbb {R})$ is given by the matrices

\begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad a,b,c,d \in \mathbb {R}, \quad ad-bc=1 .

Find the center of $SL(2,\mathbb {R})$ .

Problem 1.91

(i)(ii)

Let $A$ be an $n \times n$ matrix. Then the vec operation is defined by

\operatorname {vec}(A) := (a_{11},\ldots ,a_{n1},a_{12},\ldots ,a_{1n},\ldots ,a_{nn})^{T}

i.e. we stack the columns under each other. Consider the Pauli matrices $\sigma_{x}$ and $\sigma_{z}$

\sigma _{x} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma _{z} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} .

Then both $\left\{ \sigma_{x}, I_{2}\right\}$ and $\left\{ \sigma_{z}, I_{2}\right\}$ form a group under matrix multiplication.

(i)

Let $\otimes$ be the Kronecker product. Show that $\left\{ \sigma_{z} \otimes \sigma_{x}, I_{2} \otimes I_{2}\right\}$ form a group under matrix multiplication.

(ii)

Show that $\operatorname {vec}(\sigma_{x}) = (\sigma_{z} \otimes \sigma_{x})\operatorname {vec}(\sigma_{z})$ .

Problem 1.92

(i)(ii)

(i)

Consider the group given by the matrices

I_{2} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad U = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

and matrix multiplication. Let $x, y \in \mathbb {R}$ . Find the solutions of the equation

(xI_{2}+yU)^{2} = xI_{2}+yU .

(ii)

Using the results from (i) show that the matrices $xI_{2}+yU$ are projection matrices.

Problem 1.93

The tetrahedral group $T$ is of order $12$ . It can be generated by taking powers of products of the $3 \times 3$ matrices

g_{1} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}, \qquad g_{2} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} .

Note that $g_{2}$ is a permutation matrix. Calculate the orders of $g_{1}$ and $g_{2}$ .

Problem 1.94

(i)(ii)

Let $G$ be a finite group. For a finite group $G$ the number of conjugacy classes is equivalent to the number of non-equivalent irreducible matrix representations. Let $\left\{ C_{1},\ldots ,C_{r}\right\}$ be the set of conjugacy classes, where one sets $C_{1}$ to be the conjugacy class of $1$ , the neutral element of the finite group $G$ . Thus $C_{1} = \left\{ 1\right\}$ . Let $\left\{ \chi_{1},\ldots ,\chi_{r}\right\}$ be the set of distinct irreducible characters of $G$ . One chooses $\chi_{1}$ to be the trivial character of $G$ , i.e. $\chi_{1}(g)=1$ for all $g \in G$ . Then the character table is the $r \times r$ table $X(G) = (a_{jk})$ defined by $a_{jk} = \chi_{j}(c_{k})$ for all $j,k = 1,\ldots ,r$ , where $c_{k}$ is any element of $C_{k}$ . The value of a character is constant on each conjugacy class.

(i)

Let $\omega = e^{2\pi i/3}$ . Show that $1, \omega , \omega^{2}$ form a group under multiplication.

(ii)

Find the character table of the cyclic group $C_{3}$ .

Problem 1.95

(i)(ii)(iii)

Consider the six $3 \times 3$ permutation matrices

P_{123} = \begin{pmatrix} 1& 0& 0 \\ 0& 1& 0 \\ 0& 0& 1 \end{pmatrix}, \quad P_{132} = \begin{pmatrix} 1& 0& 0 \\ 0& 0& 1 \\ 0& 1& 0 \end{pmatrix}, \quad P_{213} = \begin{pmatrix} 0& 1& 0 \\ 1& 0& 0 \\ 0& 0& 1 \end{pmatrix},

P_{231} = \begin{pmatrix} 0& 1& 0 \\ 0& 0& 1 \\ 1& 0& 0 \end{pmatrix}, \quad P_{312} = \begin{pmatrix} 0& 0& 1 \\ 1& 0& 0 \\ 0& 1& 0 \end{pmatrix}, \quad P_{321} = \begin{pmatrix} 0& 0& 1 \\ 0& 1& 0 \\ 1& 0& 0 \end{pmatrix}

which form a group under matrix multiplication.

(i)

Find the conjugacy classes.

(ii)

Find the irreducible representations.

(iii)

Write down the character table.

Problem 1.96

(i)(ii)

The dihedral group $D_{3}$ is a non-commutative group of order $6$ . The group table is given by

\begin{array}{c|cccccc} \bullet & I & A & B & C & D & E \\ \hline I & I & A & B & C & D & E \\ A & A & B & I & D & E & C \\ B & B & I & A & E & C & D \\ C & C & E & D & I & B & A \\ D & D & C & E & A & I & B \\ E & E & D & C & B & A & I \end{array}

where $I$ is the neutral element.

(i)

Find a faithful representation by $2 \times 2$ matrices starting from the matrices

\begin{pmatrix} \cos \phi & \sin \phi \\ -\sin \phi & \cos \phi \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} \cos \phi & \sin \phi \\ \sin \phi & -\cos \phi \end{pmatrix} .

The determinant of the matrix on the left-hand side is $+1$ (which includes the identity matrix) and the determinant on the right-hand side is $-1$ .

(ii)

Find the conjugacy classes.

Problem 1.97

Consider the six-dimensional vector space $V$ consisting of polynomials of degree $2$ in two real variables $x_{1}, x_{2}$

p(x_{1},x_{2}) = c_{20}x_{1}^{2}+c_{11}x_{1}x_{2}+c_{02}x_{2}^{2}+c_{10}x_{1}+c_{01}x_{2}+c_{00}

where the $c_{jk}$ 's are real constants. Consider the faithful representation of the dihedral group given by $2 \times 2$ matrices in the previous problem. Find a six-dimensional representation of $D_{3}$ in $V$ . Let

\overrightarrow {x} = \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} \in \mathbb {R}^{2}

and $g$ be one of the $2 \times 2$ matrices of the representation. One defines

O_{g}f(\overrightarrow {x}) := f(g^{-1}\overrightarrow {x}) .

Problem 1.98

Let $A = (a_{ij})$ be a $2n \times 2n$ skew-symmetric matrix. The Pfaffian is defined as

\operatorname {Pf}(A) := \frac{1}{2^{n}n!} \sum _{\sigma \in S_{2n}} \operatorname {sgn}(\sigma ) \prod _{j=1}^{n} a_{\sigma (2j-1),\sigma (2j)}

where $S_{2n}$ is the symmetric group and $\operatorname {sgn}(\sigma )$ is the signature of permutation $\sigma$ . Consider the case with $n=2$ , i.e.

A = \begin{pmatrix} 0 & a_{12} & a_{13} & a_{14} \\ -a_{12} & 0 & a_{23} & a_{24} \\ -a_{13} & -a_{23} & 0 & a_{34} \\ -a_{14} & -a_{24} & -a_{34} & 0 \end{pmatrix} .

Calculate $\operatorname {Pf}(A)$ .

Problem 1.99

The action or realization of a group $G$ on a set $M$ is defined as a map $\varphi_{g}$

\varphi _{g} : \overrightarrow {x} \mapsto \overrightarrow {x}' = \varphi _{g}(\overrightarrow {x})

where $\overrightarrow {x}, \overrightarrow {x}' \in M$ and $g \in G$ . With each $g \in G$ a function is identified and

\varphi _{e}(\overrightarrow {x}) = x, \qquad \varphi _{g_{1}} \circ \varphi _{g_{2}} = \varphi _{g_{2}g_{1}}

for all $\overrightarrow {x} \in M$ and $g_{1}, g_{2}, e$ (identity element) $\in G$ . Let $x_{1} \sim x_{2}$ if and only if there exists $g \in G$ such that $\varphi_{g}(x_{1}) = x_{2}$ . Show that $\sim$ defines an equivalence relation.

Problem 1.100

(i)(ii)

Let $n \geq 3$ and let $\sigma_{1}, \ldots , \sigma_{n-1}$ be the generators of the braid group $\mathcal{B}_{n}$ . The braid group $\mathcal{B}_{n}$ on $n$ -strings where $n \geq 3$ has a finite presentation of $B_{n}$ given by

\left\langle \sigma _{1},\ldots ,\sigma _{n-1} : \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}, \ \sigma _{i+1}\sigma _{i}\sigma _{i+1}=\sigma _{i}\sigma _{i+1}\sigma _{i}\right\rangle

where $1 \leq i,j < n-1$ , $\left|i-j\right|>1$ or $j=n-1$ . Here $\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}$ and $\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}$ are called the braid relations. The second one is also called the Yang-Baxter equation.

(i)

Consider $B_{3}$ , $a=\sigma_{1}\sigma_{2}\sigma_{1}$ and $b=\sigma_{1}\sigma_{2}$ . Show that $a^{2}=b^{3}$ .

(ii)

Consider $B_{3}$ . The cosets $[\sigma_{1}]$ of $\sigma_{1}$ and $[\sigma_{2}]$ of $\sigma_{2}$ map to the $2 \times 2$ matrices

[\sigma _{1}] \mapsto R = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \qquad [\sigma _{2}] \mapsto L^{-1} = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}

where $L, R \in SL(2,\mathbb {Z})$ . Thus $L^{-1}, R^{-1} \in SL(2,\mathbb {Z})$ . Show that

RL^{-1}R = L^{-1}RL^{-1} .

Section 1.1Programming Problems[2/100%]

Problem 1.101

Give an efficient implementation of a group in C++ and LISP. For C++ use map<pair<G,G>,G> or map<G,map<G,G> > from the Standard Template Library. The map class in C++ implements a many to one relationship between keys and values. The map class is a template class map<keytype,valuetype> which can be used in similar way to arrays, i.e. m[key]=value. Consider for example the simplest non-trivial group $\left\{ a,b\right\}$ with group operation $\cdot$ given by $a \cdot a = b \cdot b = a$ and $a \cdot b = b \cdot a = b$ .

Problem 1.102

Consider the permutation group $S_{3}$ . Write a C++ program for the composition of the group elements using the map class of the Standard Template Library. Then implement the inverse of each group element. Finally we determine the conjugacy classes. The group consists of six elements which we denote by $a[0], a[1], \ldots , a[5]$ . The neutral (identity) element is denoted by $a[0]$ . Thus we have

a[0]*a[j] = a[j]*a[0] = a[j]

for $j=0,1,\ldots ,5$ .

Section 1.2Supplementary Problems[8/0%]

Problem 1.103

Let $A, B$ be $3 \times 3$ matrices over $\mathbb {R}$ . We define the composition

A \bullet B := A+B-2AB .

Does this composition define a group? The neutral element is $0_{3}$ .

Problem 1.104

Consider the set

G = \left\{ (a,b) \in \mathbb {R}^{2} : -\infty < a < \infty , \ 0 \leq b < 2\pi \right\} .

We define the composition $(a,b) \bullet (c,d) := (ac, (b+d) \bmod 2\pi )$ . Show that this composition defines a group.

Problem 1.105

Let $m \in \mathbb {Z}$ , $m \geq 2$ , and let $G(m)$ be the group of $2 \times 2$ matrices of determinant $\pm 1$ and with entries in the ring $\mathbb {Z}/m\mathbb {Z}$ , i.e.

G(m) := \left\{ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} : a_{11},a_{12},a_{21},a_{22} \in \mathbb {Z}/m\mathbb {Z}, \ \left|a_{11}a_{22}-a_{12}a_{21}\right|=1\right\} .

Show that the set $G(m)$ forms a finite group under matrix multiplication.

Problem 1.106

Show that there is only one group of order three. Is the group commutative? Is the group isomorphic to a subgroup of $S_{3}$ ?

Problem 1.107

Let $\phi : G \rightarrow H$ be a homomorphism between the groups $G$ and $H$ . Show that $\ker (\phi )$ is a normal subgroup of $G$ .

Problem 1.108

Let $\alpha , \beta , \phi \in \mathbb {R}$ and $\alpha , \beta \neq 0$ . Consider the matrices

A(\alpha ,\beta ,\phi ) = \begin{pmatrix} \alpha \cos \phi & -\beta \sin \phi \\ \beta ^{-1}\sin \phi & \alpha ^{-1}\cos \phi \end{pmatrix} .

Do the matrices form a group under matrix multiplication?

Problem 1.109

Let $S_{n}$ be the group of all permutations of $n$ objects. Show that every finite group is isomorphic to a group of permutations.

Problem 1.110

Let $G$ be a group. An automorphism of $G$ is an isomorphism sending $G$ onto itself. Show that the set $\operatorname {Aut}(G)$ of automorphisms of $G$ is a group with respect to the operation of composition of automorphisms.