Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 10
1. Which of the following groups is cyclic? As usual, you should prove that your
answers are correct.
(a)
(b)
(c)
(d)
Z2 × Z2
Z3 × Z2
Z5 × Z10
Z×Z
2. Let H be a subgroup of a group G, let a be a fixed element of G and consider the
set
aHa−1 = {aha−1 : h ∈ H}.
(a) Prove that θ : H → G, h 7→ aha−1 is an injective homomorphism.
(b) What can you conclude about the relationship between H and aHa−1 ?
(c) Show that depending on the choice of G, H and a, we may have aHa−1 = H
or we may have aHa−1 6= H.
3. Prove that if G, H are any groups, then the mapping θ : G → H, θ(a) = eH for
every a ∈ G is a homomorphism, and compute its kernel and image.
4. Let G and H be two groups, let θ : G → H be a homomorphism and consider the
group θ(G).
(a) Prove that if G is an abelian group, then so is θ(G).
(b) Prove that if G is a finite group, then so is θ(G). Give an example with
|θ(G)| = |G| and an example with |θ(G)| < |G|.
(c) Disprove the statement: “if G is an infinite group, then so is θ(G)”.
5. (a) Let θ : G → H be a homomorphism. Show that if J is a subgroup of H, then
K = {g ∈ G : θ(g) ∈ J}
is a subgroup of G.
(b) Apply (a) with the following data:
G = GL(2, R),
H = (R× , ·),
J = (Q ∩ (0, ∞), ·),
θ = det
to deduce that the group of 2 × 2 matrices A with real entries such that det A
is positive and rational forms a group under matrix multiplication. [Start by
explaining why J is a subgroup of H.]
6. Let G be a group. In Exercise 3 of homework exercise sheet 2, we proved that for
each x ∈ G, the mapping Lx : G → G, y 7→ xy, is a bijection, i.e. Lx ∈ Sym(G).
Consider the mapping θ : G → Sym(G), x 7→ Lx .
(a) Prove that Lx ◦ Ly = Lxy for every x, y ∈ G.
(b) Deduce that θ is an homomorphism G → (Sym(G), ◦).
(c) Compute ker θ and deduce Cayley’s theorem: every group G is isomorphic
to a permutation group [that is, a subgroup of (Sym(S), ◦), for some set S].