Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 5
1. Let S and T be subsets of the plane P = R2 . Give examples to show that both of
the statements below are false.
(a) S ⊆ T =⇒ M(S) ⊆ M(T )
(b) S ⊆ T =⇒ M(T ) ⊆ M(S)
2. Let P be the plane P = R2 . Let us say that a motion α ∈ M is written in normal
form if either α = ιP is the identity mapping, or if α is written as a composition of
at most one translation τa , at most one rotation ρθ and, perhaps, the reflection r
in the x-axis, in that order from left to right. For example, α = τa ◦ r is in
normal form, but β = r ◦ τa is not. Fill in the entries in the following Cayley
table for (M, ◦), writing your answers in normal form. [Hint: as part of this week’s
homework you’ll show that ρθ and r are linear mappings P → P .]
◦
τb
ρφ
r
τa
ρθ
r
3. Let P be the plane P = R2 .
(a) Recall that for θ ∈ R, the mapping ρθ : P → P is given by rotation by an
angle of θ radians anticlockwise about the origin. Show that ρθ ◦ ρφ = ρθ+φ
and that (ρθ )−1 = ρ−θ .
(b) Let L be a straight line in P and let rL : P → P be the mapping given by
reflection in L. If L passes through the origin and makes an angle of θ radians
with the positive x-axis, show that rL = ρ2θ ◦ r where r : P → P is reflection
in the x-axis. [Hint: one strategy is first to show that rL ◦ρ2θ ◦r is the identity
mapping using Lemma 17.]
(c) Show that if K and L are straight lines passing through the origin, then
rK ◦ rL = ρψ for some angle ψ. If K makes an angle of φ and L makes an
angle of θ with the positive x-axis, find ψ in terms of φ and θ.
[Hint: first show that ρt ◦ r = r ◦ ρ−t by considering (ρt ◦ r)−1 .]
4. Let (G, ∗) be a group with identity element e, and let x ∈ G. If xk = e for some
k ∈ N, then the order o(x) of x is the smallest k ∈ N such that xk = e. If xk 6= e
for every k ∈ N, then o(x) = ∞.
Consider the dihedral group
D12 = M(T ) = {e, ρ, ρ2 , . . . , ρ11 , r0 , r1 , . . . , r11 }
where T is the regular polygon with
12 sides inscribed in the unit circle centred at
the origin with one vertex at 10 , e is the identity mapping on P , ρ = ρπ/6 and
rm is reflection in the line through the origin making an angle mπ/12 with the
positive x-axis.
Find o(α) for each α ∈ D12 .
5. For which integers n, m ≥ 3 is Dn a subgroup of (Dm , ◦)?