5) Group Actions Flashcards
What is a group action
What is an orbit
What is the relationship between the orbits of two elements in a group action
Let G be a group acting on X, and x, y ∈ X. Then x ∈ Orb(y) if and only if Orb(x) = Orb(y)
How is the equivalence of orbits proved in a group action
How do orbits from a group action on a set partition the set
Let G be a group acting on a set X. Then X is the disjoint union of the different orbits
How is it proven that orbits from a group action partition a set
What is a Transitive action
The action of G on X is called transitive if X is a single orbit, in other words, if for all x, y ∈ X there exists g ∈ G with g.x = y
What is a Fixed points
An element x ∈ X is called a fixed point of the action of G on X if its orbit has size 1, in other words, if g.x = x for all g ∈ G
What are Stabilisers
Let G be a group acting on a set X. The stabiliser of a
point x ∈ X is the set Stab(x) = {g ∈ G | g.x = x} ⊆ G
Why is the stabiliser of an element in a set action considered a subgroup
What is The Orbit-Stabiliser Theorem
Describe the proof of the Orbit-Stabiliser Theorem
Does the Orbit-Stabiliser Theorem apply to infinite groups and sets
- The relationship ∣Orb(x)∣=[G:Stab(x)] holds true even if the group G and the set X are infinite. The proof constructs a bijection between the orbit of x and the set of left cosets of Stab(x) without relying on finiteness
- In infinite contexts, direct division might not hold conceptual or practical meaning since we don’t know what this division means if |Stab(x)| is infinite
How does the size of an orbit relate to the order of a group in finite contexts
If G is a finite group acting on a finite set X, then the size of every orbit divides the order of G.
Are all orbits the same size
No, orbits can differ in size
