Physics 5040
Spring 2009
Problem Set 8
These problems cover the entire set of group theory notes, including the parts I
didn’t cover in lecture.
1. Decide which of the following sets G forms a group under the indicated operation. If G does not form a group, give the reason.
(a) G = {all integers} under ordinary subtraction.
(b) G = {all nonzero rational numbers} under ordinary division.
(c) G = {a0 , a1 , . . . , a6 } where
(
ai+j
if i + j < 7
ai aj =
ai+j−7 if i + j ≥ 7
(d) G = {2m 3n : m, n ∈ Z} under ordinary multiplication.
2. Let F [R] be the set of all real-valued functions of a real variable. As usual,
for any f, g ∈ F [R] we define (f + g)(x) = f (x) + g(x) for each x ∈ R, so that
f + g ∈ F [R]. Show that F [R] forms a group.
3. (a) Given two groups A and B, we can form the Cartesian product A × B =
{(a, b) : a ∈ A and b ∈ B} of these groups considered as sets. Prove that
A × B can be made into a group with respect to the operation defined by
(a1 , b1 )(a2 , b2 ) = (a1 a2 , b1 b2 ) for all a1 , a2 ∈ A and b1 , b2 ∈ B. This group
is called the direct product of A and B.
(b) Prove that {(x, x) : x ∈ G} is a subgroup of G × G. This is called the
diagonal subgroup of G × G.
4. Express each of the following as the product of disjoint cycles:
(a) (1, 2, 3)(4, 5)(1, 6, 7, 8, 9)(1, 5)
(b) (1, 2)(1, 2, 3)(1, 2)
5. (a) Show that the composition φ ◦ ψ : A → C is a homomorphism if both
φ : B → C and ψ : A → B are.
(b) Show that the set of all group isomorphisms defines an equivalence relation
on the set of all groups.
6. Determine which of the following mappings φ : G → G′ are homomorphisms,
and for those that are, determine their kernel:
(a) G = G′ = the group of nonzero real numbers under multiplication, and
φ(x) = x2 for all x ∈ G.
(b) G = G′ = the group of all real numbers under addition, and φ(x) = 1 + x
for all x ∈ G.
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(c) Repeat part (b), but with φ(x) = kx for any (fixed) number k.
7. (a) Let φ be a homomorphism of a group G into a group G′ , and let Kφ be
the kernel of φ. Prove that Kφ is a normal subgroup of G.
(b) The center of a group G is the set Z = {z ∈ G : zg = gz for all g ∈ G}.
Show that Z is a normal subgroup of G.
8. The simplest non-cyclic group is of order 4, and is called the dihedral group
D2 . This is the group of symmetry transformations of the rectangle:
b
a
Besides the identity e, the group consists of the elements {a, b} that are rotations by π about the axes shown, plus the element {c} that is a rotation by π
about an axis perpendicular to the page.
(a) Fill in the group multiplication table:
e
a
b
c
e
a
b
c
(b) Find the subgroup of S4 isomorphic to D2 .
9. Prove that if a representation D(G) is reducible, then so is the representation
U (G) = SD(G)S −1 .
10. Referring to the discussion following Example 18 of the notes, let Ci and Cj
be two classes of a group G, and let Ci′ be the class of elements inverse to
those of Ci . If j 6= i′ , show that Ci Cj can not contain C1 = [e].
11. (a) Prove the following theorem:
Let φ be a homomorphism of a group G onto a group G′ , and let Kφ be
the kernel of φ. By Exercise 7a, we know that Kφ is a normal subgroup of
G, and hence we may form the quotient group G/Kφ . Prove that G/Kφ
is isomorphic to G′ .
Hint : Since any element in X ∈ G/Kφ is of the form Kφ g where g ∈ G,
define the mapping ψ : G/Kφ → G′ by ψ(X) = ψ(Kφ g) = φ(g). To
show that ψ is an isomorphism, first show that ψ is well-defined, that is,
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X = Kφ g = Kφ g ′ implies φ(g) = φ(g ′ ). Next, show that ψ is a homomorphism, i.e., that ψ(XY ) = ψ(X)ψ(Y ). Now show that ψ is surjective
(use the fact that φ is surjective). Finally, show that Ker ψ = {0} (you
will need the additional fact that the identity in G/Kφ is Kφ = Kφ e).
(b) Prove the converse part of Theorem 11 in the notes. Hint : The fact that
U (G) is degenerate means that its kernel is non-trivial.
12. Find all the inequivalent irreducible representations (or at least their dimension) of S3 , and construct its character table.
Hints: Here is what you should know from theorems and examples in the
notes:
(i) The number of inequivalent irreps.
(ii) There is a trivial identity rep D1 (G).
(iii) There is a second one-dimensional irrep D2 (G) that follows from Example 21.
(iv) Now what is the dimension of the next irrep?
(v) Now use the orthogonality relations for the characters.
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