Exercises

GROUP THEORY
mathematical details, a rotation about axis i can be represented by the operator
Ri (θ), and the two rotations are connected by a relationship of the form
Rj (θ) = φ−1
ij Ri (θ)φij ,
in which φij is the member of the full continuous rotation group that takes axis
i into axis j.
28.8 Exercises
28.1
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)
(b)
(c)
(d)
(e)
the integers
the integers
the integers
the integers
all matrices
(mod 10) under addition;
(mod 10) under multiplication;
1, 2, 3, 4, 5, 6 under multiplication (mod 7);
1, 2, 3, 4, 5 under multiplication (mod 6);
of the form
a a−b
,
0
b
where a and b are integers (mod 5) and a = 0 = 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


1 0 0
 a 1 0 ,
b c 1
where a, b, c are integers, under matrix multiplication.
28.2
Which of the following relationships between X and Y are equivalence relations?
Give a proof of your conclusions in each case:
(a)
(b)
(c)
(d)
(e)
(f)
28.3
X and Y are integers and X − Y is odd;
X and Y are integers and X − Y is even;
X and Y are people and have the same postcode;
X and Y are people and have a parent in common;
X and Y are people and have the same mother;
X and Y are n×n matrices satisfying Y = P XQ, where P and Q are elements
of a group G of n × n matrices.
Define a binary operation • on the set of real numbers by
x • y = x + y + rxy,
where r is a non-zero real number. Show that the operation • is associative.
Prove that x • 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 •.
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28.8 EXERCISES
28.4
28.5
Prove that the relationship X ∼ Y , defined by X ∼ Y if Y can be expressed in
the form
aX + b
Y =
,
cX + d
with a, b, c and d as integers, is an equivalence relation on the set of real numbers
. Identify the class that contains the real number 1.
The following is a ‘proof’ that reflexivity is an unnecessary axiom for an equivalence relation.
Because of symmetry X ∼ Y implies Y ∼ X. Then by transitivity X ∼ Y and
Y ∼ X imply X ∼ 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 ∼ Y if X + Y is real.
(b) X ∼ Y if XY is real.
28.6
Prove that the set M of matrices
A=
a
0
b
c
,
where a, b, c are integers (mod 5) and a = 0 = c, form a non-Abelian group
under matrix multiplication.
Show that the subset containing elements of M that are of order 1 or 2 do
not form a proper subgroup of M,
(a) using Lagrange’s theorem,
(b) by direct demonstration that the set is not closed.
28.7
28.8
28.9
28.10
S is the set of all 2 × 2 matrices of the form
w x
A=
,
where wz − xy = 1.
y z
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.
Show that, under matrix multiplication, matrices of the form
a0 + a1 i −a2 + a3 i
M(a0 , a) =
,
a2 + a3 i
a0 − a1 i
where a0 and the components of column matrix a = (a1 a2 a3 )T are real
numbers satisfying a20 + |a|2 = 1, constitute a group. Deduce that, under the
transformation z → Mz, where z is any column matrix, |z|2 is invariant.
If A is a group in which every element other than the identity, I, has order 2,
prove that A is Abelian. Hence show that if X and Y are distinct elements of A,
neither being equal to the identity, then the set {I, X, Y , XY } forms a subgroup
of A.
Deduce that if B is a group of order 2p, with p a prime greater than 2, then B
must contain an element of order p.
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.
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GROUP THEORY
(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’, say, and a second diagonal made to
lie along a particular direction.
28.11
28.12
Identify the eight symmetry operations on a square. Show that they form a
group D4 (known to crystallographers as 4mm and to chemists as C4v ) having one
element of order 1, five of order 2 and two of order 4. Find its proper subgroups
and the corresponding cosets.
If A and B are two groups, then their direct product, A × B, is defined to be
the set of ordered pairs (X, Y ), with X an element of A, Y an element of B
and multiplication given by (X, Y )(X , Y ) = (XX , Y Y ). Prove that A × B is a
group.
Denote the cyclic group of order n by Cn and the symmetry group of a regular
n-sided figure (an n-gon) by Dn – thus D3 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 C2 × C3 is
isomorphic to C6 , and (ii) that C2 × D3 is isomorphic to D6 .
(b) Are any of D4 , C8 , C2 × C4 , C2 × C2 × C2 isomorphic?
28.13
28.14
28.15
Find the group G generated under matrix multiplication by the matrices
0 1
0 i
A=
,
B=
.
1 0
i 0
Determine its proper subgroups, and verify for each of them that its cosets
exhaust G.
Show that if p is prime then the set of rational number pairs (a, b), excluding
(0, 0), with multiplication defined by
√
√
√
(a, b) • (c, d) = (e, f), where (a + b p)(c + d p) = e + f p,
forms an Abelian group. Show further that the mapping (a, b) → (a, −b) is an
automorphism.
Consider the following mappings between a permutation group and a cyclic
group.
(a) Denote by An the subset of the permutation group Sn that contains all the
even permutations. Show that An is a subgroup of Sn .
(b) List the elements of S3 in cycle notation and identify the subgroup A3 .
(c) For each element X of S3 , let p(X) = 1 if X belongs to A3 and p(X) = −1 if it
does not. Denote by C2 the multiplicative cyclic group of order 2. Determine
the images of each of the elements of S3 for the following four mappings:
Φ1 : S3 → C2
Φ2 : S3 → C2
Φ3 : S3 → A3
Φ4 : S3 → S3
X
X
X
X
→ p(X),
→ −p(X),
→ X 2,
→ X 3.
(d) For each mapping, determine whether the kernel K is a subgroup of S3 and,
if so, whether the mapping is a homomorphism.
28.16
For the group G with multiplication table 28.8 and proper subgroup H = {I, A, B},
denote the coset {I, A, B} by C1 and the coset {C, D, E} by C2 . Form the set of
all possible products of a member of C1 with itself, and denote this by C1 C1 .
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28.8 EXERCISES
28.17
28.18
28.19
Similarly compute C2 C2 , C1 C2 and C2 C1 . Show that each product coset is equal to
C1 or to C2 , and that a 2 × 2 multiplication table can be formed, demonstrating
that C1 and C2 are themselves the elements of a group of order 2. A subgroup
like H whose cosets themselves form a group is a normal subgroup.
The group of all non-singular n × n matrices is known as the general linear
group GL(n) and that with only real elements as GL(n, R). If R∗ denotes the
multiplicative group of non-zero real numbers, prove that the mapping Φ :
GL(n, R) → R∗ , defined by Φ(M) = det M, is a homomorphism.
Show that the kernel K of Φ is a subgroup of GL(n, R). Determine its cosets
and show that they themselves form a group.
The group of reflection–rotation symmetries of a square is known as D4 ; let
X be one of its elements. Consider a mapping Φ : D4 → S4 , the permutation
group on four objects, defined by Φ(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
the two principal diagonals, of the square. For example, if R is a rotation by
π/2, Φ(R) = (12)(34). Show that D4 is mapped onto a subgroup of S4 and, by
constructing the multiplication tables for D4 and the subgroup, prove that the
mapping is a homomorphism.
Given that matrix M is a member of the multiplicative group GL(3, R), determine,
for each of the following additional constraints on M (applied separately), whether
the subset satisfying the constraint is a subgroup of GL(3, R):
(a)
(b)
(c)
(d)
28.20
MT = M;
MT M = I;
|M| = 1;
Mij = 0 for j > i and Mii = 0.
The elements of the quaternion group, Q, are the set
{1, −1, i, −i, j, −j, k, −k},
with i = j = k = −1, ij = k and its cyclic permutations, and ji = −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 4mm
(C4v ) considered in exercise 28.11.
Show that D4 , 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 Q, of exercise 28.20, onto one (and hence either) of these two
subgroups, and determine its kernel.
Show that the matrices


cos θ − sin θ x

sin θ
cos θ
y ,
M(θ, x, y) =
0
0
1
2
28.21
28.22
28.23
2
2
where 0 ≤ θ < 2π, −∞ < x < ∞, −∞ < y < ∞, form a group under matrix
multiplication.
Show that those M(θ, x, y) for which θ = 0 form a subgroup and identify its
cosets. Show that the cosets themselves form a group.
Find (a) all the proper subgroups and (b) all the conjugacy classes of the symmetry
group of a regular pentagon.
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