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Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 24

Group theory - all with Video Answers

Educators

Chapter Questions

07:19

Problem 1

For each of the following sets, determine whether they form a group under the operation indicated (where it is relevant you may assume that matrix multiplication is associative):
(a) the integers (mod 10 ) under addition;
(b) the integers (mod 10 ) under multiplication;
(c) the integers $1,2,3,4,5,6$ under multiplication (mod 7 );
(d) the integers $1,2,3,4,5$ under multiplication (mod 6 );
(e) all matrices of the form
$$
\left(\begin{array}{cc}
a & a-b \\
0 & b
\end{array}\right)
$$
where $a$ and $b$ are integers (mod 5 ), and $a \neq 0 \neq b$, under matrix multiplication;
(f) those elements of the set in (e) that are of order 1 or 2 (taken together);
(g) all matrices of the form
$\left(\begin{array}{lll}1 & 0 & 0 \\ a & 1 & 0 \\ b & c & 1\end{array}\right) \quad$ where $a, b, c$ are integers,
under matrix multiolication

Ely Crowder
Ely Crowder
Numerade Educator
09:20

Problem 2

Which of the following relationships between $X$ and $Y$ are equivalence relations? Give a proof of your conclusions in each case:
(a) $X$ and $Y$ are integers and $X-Y$ is odd;
(b) $X$ and $Y$ are integers and $X-Y$ is even;
(c) $X$ and $Y$ are people and have the same postcode,
(d) $X$ and $Y$ are people and have a parent in common;
(e) $X$ and $Y$ are people and have the same mother;
(f) $X$ and $Y$ are $n \times n$ matrices satisfying $Y=P X Q$, where $P$ and $Q$ are elements of a group $\mathcal{G}$ of $n \times n$ matrices.

Ely Crowder
Ely Crowder
Numerade Educator
08:33

Problem 3

Define a binary operation $\bullet$ on the set of real numbers by
$$
x \bullet y=x+y+r x y
$$
where $r$ is a non-zero real number. Show that the operation $\bullet$ is associative. Prove that $x \bullet y=-r^{-1}$ if, and only if, $x=-r^{-1}$ or $y=-r^{-1}$. Hence prove that the set of all real numbers excluding $-r^{-1}$ forms a group under the operation $\bullet$.

Ely Crowder
Ely Crowder
Numerade Educator
05:09

Problem 4

Prove that the relationship $X \sim Y$, defined by $X \sim Y$ if $Y$ can be expressed in the form
$$
Y=\frac{a X+b}{c X+d}
$$
with $a, b, c$ and $d$ as integers, is an equivalence relation on the set of real numbers R. Identify the class that contains the real number $1 .$

Mj Santos
Mj Santos
Numerade Educator
07:03

Problem 5

The following is a 'proof' that reflexivity is an unnecessary axiom for an equivalence relation. Because of symmetry $X \sim Y$ implies $Y \sim X .$ Then by transitivity $X \sim Y$ and $Y \sim X$ imply $X \sim X .$ Thus symmetry and transitivity imply reflexivity, which therefore need not be separately required.
Demonstrate the flaw in this proof using the set consisting of all real numbers plus the number $i .$ Show by investigating the following specific cases that, whether or not reflexivity actually holds, it cannot be deduced from symmetry and transitivity alone.
(a) $X \sim Y$ if $X+Y$ is real.
(b) $X \sim Y$ if $X Y$ is real.

Ely Crowder
Ely Crowder
Numerade Educator
View

Problem 6

Prove that the set $\mathcal{M}$ of matrices
$$
A=\left(\begin{array}{ll}
a & b \\
0 & c
\end{array}\right)
$$
where $a, b, c$ are integers (mod 5 ) and $a \neq 0 \neq c$, forms a non-Abelian group under matrix multiplication.

Show that the subset containing elements of $M$ that are of order 1 or 2 does not form a proper subgroup of $\mathcal{M}$
(a) using Lagrange's theorem,
(b) by direct demonstration that the set is not closed.

Nick Johnson
Nick Johnson
Numerade Educator
11:18

Problem 7

$S$ is the set of all $2 \times 2$ matrices of the form
$$
A=\left(\begin{array}{cc}
w & x \\
y & z
\end{array}\right) \quad \text { where } w z-x y=1
$$
Show that $S$ is a group under matrix multiplication. Which element(s) have order 2? Prove that an element $A$ has order 3 if $w+z+1=0$

Ely Crowder
Ely Crowder
Numerade Educator
01:08

Problem 8

Show that, under matrix multiplication, matrices of the form
$$
\mathrm{M}\left(a_{0}, \mathrm{a}\right)=\left(\begin{array}{cc}
a_{0}+a_{1} i & -a_{2}+a_{3} \hat{i} \\
a_{2}+a_{3} i & a_{0}-a_{1} i
\end{array}\right)
$$
where $a_{0}$ and the components of column matrix $\mathrm{a}=\left(\begin{array}{lll}a_{1} & a_{2} & a_{3}\end{array}\right)^{\mathrm{T}}$ are real numbers satisfying $a_{0}^{2}+|\mathrm{a}|^{2}=1$, form a group. Deduce that, under the transformation $z \rightarrow M z$, where $z$ is anv column matrix, $|z|^{2}$ is invariant

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
06:02

Problem 9

If $\mathcal{A}$ is a group in which every element other than the identity, $I$, has order 2, prove that $\mathcal{A}$ is Abelian. Hence show that if $X$ and $Y$ are distinct elements of $\mathcal{A}$, neither being equal to the identity, then the set $\{I, X, Y, X Y\}$ forms a subgroup of $\mathcal{A}$.

Deduce that if $\mathcal{B}$ is a group of order $2 p$, with $p$ a prime greater than 2, then $\mathcal{B}$ must contain an element of order $p .$

Ely Crowder
Ely Crowder
Numerade Educator
03:42

Problem 10

The group of rotations (excluding reflections and inversions) in three dimensions that take a cube into itself is known as the group 432 (or $O$ in the usual chemical notation). Show by each of the following methods that this group has 24 elements.
(a) Identify the distinct relevant axes and count the number of qualifying rotations about each.
(b) The orientation of the cube is determined if the directions of two of its body diagonals are given. Consider the number of distinct ways in which one body diagonal can be chosen to be 'vertical' and a second diagonal made to lie along a particular direction.

Dominador Tan
Dominador Tan
Numerade Educator
08:20

Problem 11

Identify the eight symmetry operations on a square. Show that they form a group (known to crystallographers as $4 \mathrm{~mm}$ or to chemists as $C_{4 c}$ ) having one element of order 1, five of order 2 and two of order $4 .$ Find its proper subgroups and the corresponding cosets.

Ely Crowder
Ely Crowder
Numerade Educator
06:02

Problem 12

If $\mathcal{A}$ and $\mathcal{B}$ are two groups then their direct product, $A \times \mathcal{B}$, is defined to be the set of ordered pairs $(X, Y)$, with $X$ an element of $\mathcal{A}, Y$ an element of $\mathcal{B}$ and multiplication given by $(X, Y)\left(X^{\prime}, Y^{\prime}\right)=\left(X X^{\prime}, Y Y^{\prime}\right)$. Prove that $\mathcal{A} \times \mathcal{B}$ is a group.

Denote the cyclic group of order $n$ by $\mathcal{C}_{n}$ and the symmetry group of a regular $n$-sided figure (an $n$-gon) by $\mathcal{D}_{n}-$ thus $\mathcal{D}_{3}$ is the symmetry group of an equilateral triangle, as discussed in the text.
(a) By considering the orders of each of their elements, show (i) that $\mathcal{C}_{2} \times \mathcal{C}_{3}$ is isomorphic to $\mathcal{C}_{6}$, and (ii) that $\mathcal{C}_{2} \times \mathcal{D}_{3}$ is isomorphic to $\mathcal{D}_{6}$.
(b) Are any of $\mathcal{D}_{4}, \mathcal{C}_{8}, \mathcal{C}_{2} \times \mathcal{C}_{4}, \mathcal{C}_{2} \times \mathcal{C}_{2} \times \mathcal{C}_{2}$ isomorphic?

Ely Crowder
Ely Crowder
Numerade Educator
11:16

Problem 13

Find the group $\mathcal{G}$ generated under matrix multiplication by the matrices
$$
\mathrm{A}=\left(\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right), \quad \mathrm{B}=\left(\begin{array}{ll}
0 & i \\
i & 0
\end{array}\right)
$$
Determine its proper subgroups, and verify for each of them that its cosets exhaust $\mathcal{G}$.

Ely Crowder
Ely Crowder
Numerade Educator
03:38

Problem 14

Show that if $p$ is prime then the set of rational number pairs $(a, b)$, excluding $(0,0)$, with multiplication defined by
$$
(a, b) \cdot(c, d)=(e, f), \quad \text { where }(a+b \sqrt{p})(c+d \sqrt{p})=e+f \sqrt{p}
$$
forms an Abelian group. Show further that the mapping $(a, b) \rightarrow(a,-b)$ is an automornhism

Wendi Zhao
Wendi Zhao
Numerade Educator
01:40

Problem 15

(a) Denote by $A_{n}$ the subset of the permutation group $S_{n}$ that contains all the even permutations. Show that $A_{n}$ is a subgroup of $S_{n}$.
(b) List the elements of $S_{3}$ in cycle notation and identify the subgroup $A_{3} .$
(c) For each element $X$ of $S_{3}$, let $p(X)=1$ if $X$ belongs to $A_{3}$ and $p(X)=-1$ if it does not. Denote by $\mathcal{C}_{2}$ the multiplicative cyclic group of order 2 . Determine the images of each of the elements of $S_{3}$ for the following four mappings:
$$
\begin{array}{ll}
\Phi_{1}: S_{3} \rightarrow \mathcal{C}_{2} & X \rightarrow p(X) \\
\Phi_{2}: S_{3} \rightarrow \mathcal{C}_{2} & X \rightarrow-p(X) \\
\Phi_{3}: S_{3} \rightarrow A_{3} & X \rightarrow X^{2} \\
\Phi_{4}: S_{3} \rightarrow S_{3} & X \rightarrow X^{3}
\end{array}
$$
(d) For each mapping, determine whether the kernel $K$ is a subgroup of $S_{3}$ and, if so, whether the mapping is a homomorphism.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 16

For the group $\mathcal{G}$ with multiplication table $24.8$ and proper subgroup $\mathcal{H}=\{I, A, B\}$, denote the coset $\{I, A, B\}$ by $\mathcal{C}_{1}$ and the coset $\{C, D, E\}$ by $\mathcal{C}_{2}$. Form the set of all possible products of a member of $\mathcal{C}_{1}$ with itself, and denote this by $\mathcal{C}_{1} \mathcal{C}_{1}$. Similarly compute $\mathcal{C}_{2} \mathcal{C}_{2}, \mathcal{C}_{1} \mathcal{C}_{2}$ and $\mathcal{C}_{2} \mathcal{C}_{1}$. Show that each product coset is equal to $\mathcal{C}_{1}$ or to $\mathcal{C}_{2}$ and that a $2 \times 2$ multiplication table can be formed demonstrating that $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are themselves the elements of a group of order $2 .$ A subgroup like $\mathcal{H}$ whose cosets themselves form a group is a normal subgroup.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:25

Problem 17

The group of all non-singular $n \times n$ matrices is known as the general linear group $G L(n)$ and that with only real elements as $G L(n, \mathbf{R}) .$ If $\mathbf{R}^{*}$ denotes the multiplicative group of non-zero real numbers, prove that the mapping $\Phi$ : $G L(n, \mathbf{R}) \rightarrow \mathbf{R}^{*}$, defined by $\Phi(\mathrm{M})=\operatorname{det} \mathrm{M}$, is a homomorphism.

Show that the kernel $\mathcal{K}$ of $\Phi$ is a subgroup of $G L(n, \mathbf{R})$. Determine its cosets and show that they themselves form a group.

Ely Crowder
Ely Crowder
Numerade Educator
05:10

Problem 18

The group of reflection-rotation symmetries of a square is known as $\mathcal{D}_{4} ;$ let $X$ be one of its elements. Consider a mapping $\Phi: \mathcal{D}_{4} \rightarrow S_{4}$, the permutation group on four objects, defined by $\Phi(X)=$ the permutation induced by $X$ on the set $\{x, y, d, d\}$, where $x$ and $y$ are the two principal axes and $d$ and $d^{\prime}$ the two principal diagonals, of the square. For example, if $R$ is a rotation by $\pi / 2, \Phi(R)=(12)(34)$. Show that $\mathcal{D}_{4}$ is mapped onto a subgroup of $S_{4}$ and, by constructing the multiplication tables for $\mathcal{D}_{4}$ and the subgroup, prove that the mapping is a homomorphism.

Ely Crowder
Ely Crowder
Numerade Educator
05:29

Problem 19

Given that matrix $M$ is a member of the multiplicative group $G L(3, \mathbf{R})$, determine, for each of the following additional constraints on $M$ (applied separately), whether the subset satisfying the constraint is a subgroup of $G L(3, \mathbf{R})$ :
(a) $\mathrm{M}^{T}=\mathrm{M}$
(b) $\mathrm{M}^{\bar{T}} \mathrm{M}=\mathrm{I}$;
(c) $|\mathrm{M}|=1$;

Ely Crowder
Ely Crowder
Numerade Educator
05:10

Problem 20

In the quaternion group $Q$ the elements form the set
$$
\{1,-1, i,-i, j,-j, k,-k\}
$$
with $i^{2}=j^{2}=k^{2}=-1, i j=k$ and its cyclic permutations, and $j i=-k$ and its cyclic permutations. Find the proper subgroups of $Q$ and the corresponding cosets. Show that the subgroup of order 2 is a normal subgroup, but that the other subgroups are not. Show that $Q$ cannot be isomorphic to the group $4 m m$ $\left(C_{4 c}\right)$ considered in exercise $24.11 .$

Ely Crowder
Ely Crowder
Numerade Educator
08:50

Problem 21

Show that $\mathcal{D}_{4}$, the group of symmetries of a square, has two isomorphic subgroups of order 4 . Show further that there exists a two-to-one homomorphism from the quaternion group 2 of exercise $24.20$ onto one (and hence either) of these two subgroups, and determine its kernel.

Ely Crowder
Ely Crowder
Numerade Educator
02:46

Problem 22

(a) No, not reflexive; (b) yes, partition of integers into odd and even; (c) yes. (d) no, not transitive, $X \rightarrow Y \rightarrow Z$ if $Y^{\prime}$ s parents both re-marry and $X$ and $Z$ are children of the two second marriages; (e) yes; (f) yes.

Srilakshmi E K
Srilakshmi E K
Numerade Educator
06:41

Problem 23

Find (a) all the proper subgroups and (b) all the conjugacy classes of the symmetry group of a regular pentagon.

Ely Crowder
Ely Crowder
Numerade Educator