Mathematics 1214: Introduction to Group Theory
Exercise sheet 11
1. Let n ≥ 3 and let Dn be the dihedral group of order 2n. Writing ρ = ρ2π/n , show
that hρi is a normal subgroup of Dn . [Hint: if r is reflection in the x-axis, then
r ◦ ρ = ρ−1 ◦ r, and the reflections in Dn are rj = ρj ◦ r for 0 ≤ j < n.] What is
the order of the quotient group Dn /hρi?
2. Give an example of a normal subgroup of GL(2, R), and give an example of a
subgroup of GL(2, R) which is not normal.
3. Let G be a group and let N be a normal subgroup of G.
(a) Prove that the mapping η : G → G/N defined by η(a) = N a for a ∈ G is a
homomorphism. What are the kernel and image of η?
(b) If H is another group and θ : G → H is a homomorphism with kernel N ,
and ϕ : G/N → H is defined by ϕ(N a) = θ(a), then ϕ is a well-defined
homomorphism (see the proof of the Fundamental Homomorphism Theorem).
Prove that θ = ϕ ◦ η.
4. Let θ : R → C× , θ(x) = eix . Prove that θ is a homomorphism with kernel h2πi,
and that R/h2πi ≈ {z ∈ C× : |z| = 1}.
5. Let G be a group with identity element e.
(a) Show that {e} ⊳ G and G/{e} ≈ G.
(b) Show that G ⊳ G and G/G ≈ {e}.
6. Let G be a group.
(a) Show that if N is a normal subgroup of G, then G/N is an abelian group if
and only if aba−1 b−1 ∈ N for all a, b ∈ G.
(b) Give an example of a non-abelian group G and a normal subgroup N of G
such that G/N is a finite abelian group. [You can always do this easily with
N = G. So if you want a little more of a challenge, try to do it with N 6= G.]
(c) Give another example of a non-abelian group G and a normal subgroup N
of G such that G/N is an infinite abelian group.
7. If G is a cyclic group and N is a subgroup of G, explain why N is a normal
subgroup of G and prove that G/N is a cyclic group. [Hint: for the second part,
if G = hai, prove that G/N = hN ai.]
8. Let G be a group and let N ⊳ G.
(a) If G is a finite group, prove that G/N is a finite group.
(b) If G is an infinite group and N is a finite normal subgroup, prove that G/N
is an infinite group.
(c) Show (by giving examples) that if G is an infinite group and N is an infinite
normal subgroup, then G/N may be finite, or it may be infinite.