4. Group Actions

Question 1

4.10.
Give the definition of the Orbit of x under the action of G.

Answer 1

Definition 4.10.
Let G be a group acting on a set X and x ∈ X. We call
orbG(x)={g▷x : g∈G}
Any subset O ⊂ X such that O = orbG(x) for some x ∈ X is called an orbit under the action of G.

Question 2

4.12
When is the action of a group G on a set X called Transitive?

Answer 2

Defintion 4.12
When there only exists one orbit, in other words if orbG(x) = X for all x ∈ X.

Question 3

4.15
Give the definition of a Stabiliser of x in G.

Answer 3

Defintion 4.15
Let G be a group acting on a set X and x ∈X. We call the set
stabG(x)={g∈G : g▷x=x}
the Stabiliser of x in G.

Question 4

Let G be a group. How do the amount of Stabilisers present, the number of Orbits and the order of the group relate.

Answer 4

Orbits x Stabilisers = Number of elements in the group