Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 6
1. Let S be a non-empty set, and let EquivRels(S) be the set of all equivalence
relations on S, and let Partitions(S) be the set of all partitions of S.
(a) Given an equivalence relation ∼ in EquivRels(S), let α(∼) denote the collection of equivalence classes of ∼. Explain why α(∼) ∈ Partitions(S).
(b) Given a partition P in Partitions(S), let β(P) be the relation on S defined
by
x β(P) y ⇐⇒ ∃ A ∈ P : x ∈ A and y ∈ A.
Explain why β(P) ∈ EquivRels(S).
(c) By (a) and (b), we have two well-defined mappings,
α : EquivRels(S) → Partitions(S) and β : Partitions(S) → EquivRels(S).
Prove that these mappings are inverses of one another.
(d) Deduce that α and β are both bijections.
2. In Theorem 20, we showed that if G is a permutation group on a set S, then the
relation ∼ on S defined by
x ∼ y ⇐⇒ ∃ α ∈ G : α(x) = y
is an equivalence relation on S, called G-orbit equivalence. If x ∈ S then we call
the equivalence class [x]∼ the G-orbit of x.
For each of the following sets G, (i) show that G is a permutation group on the
plane P , (ii) compute the G-orbits and (iii) find a complete set of equivalence class
representatives for G-orbit equivalence.
(a) G = {τa : a ∈ P }
(b) G = {τa : a =
x
y
, x, y ∈ Z}
(c) G = {ρθ : θ ∈ R}
(d) G = MT where T = {
(e) G = M(T ) where T = {
0
0
}
x
0
: x ∈ R}
(f) G = {ιP , r}
(g) G = D4
[Use the partition {(a), (b), (c)}, {(d), (e)}, {(f), (g)} for groupwork purposes]