Chapter 6-Group Actions And Permutation Groups

Question 1

For a finite set Ω, define a permutation of Ω and the group of permutations of Ω. What is a permutation group?

Answer 1

A permutation of Ω is a bijection. The group of perms is SΩ, written Sn if Ω is the set of integers up to n. a permutation group of degree m is a pair (G, Ω) where G=<SΩ
and |Ω|=m.

Question 2

Define the kernel of an action.
When is an action faithful?
If we have a permutation group what is the natural action?
Define the coset action and the right regular action. What is the kernel if the coset action?

Answer 2

If G acts on Ω then the kernel of the action is:
G(Ω)={g in G : αg=α for all α in Ω}
An action is faithful if the kernel is trivial. Ie if all elements in G move at least some point in Ω.
Let (G,Ω) be a permutation group. Then G acts faithfully on Ω. This is called the natural action.
If H=<G we have a natural action on the set of right cosets of H in G. Namely:
Hx•g=Hxg
This is the coset action. When H=1 we have G acting on itself and this is then the right regular action.
If g is in the kernel of the coset action then g is in the intersection of x^-1Hx for all x, called the core of H in G.

Question 3

What is the orbit of an element under an action of G on Ω?
What is the stabiliser?
Show that G(αg)=g^-1(Gα)g

Answer 3

The orbit of α in Ω is:
αG={αg : g in G}
The stabiliser of α is:
Gα={g in G : αg=α}

x is in G(αg) IFF αgx=αg IFF gxg^-1 is in Gα IFF x is in the final bit.

Question 4

When are two actions of G on different sets equivalent?
Prove that if G acts on Ω an α €Ω then the actions of G on αG and (G:Gα) are equivalent - ie the action of G on the orbit of an element is equivalent to the action of G on the cosets of the stabiliser of the same element.

Answer 4

If G acts on Ω, Ω’ then they are equivalent if there is a bijection:
Θ : Ω –> Ω’
Such that for all g in G and α in Ω, we have (αΘ)g = (αg)Θ

Proof: given h, g in G we have
αg=αh IFF g•h^-1 is in Gα
IFF (Gα)g=(Gα)h
So the map αg |–> (Gα)g is a well defined bijection with required properties.

Question 5

Define the fixed point set of g for some element in G when G acts on Ω.
When is g fixed point free?

Answer 5

This is fix(g)={α in Ω : αg=α}

g is fixed point free if the fixed point set of g is empty.

Question 6

Show that if G acts on Ω then the number of orbits is:

(1/|G|)•Σ|fix(g)|
Where we sum over all g.

Prove the corollary that if G acts transitively then some element is completely fixed point free

Answer 6

Proof:

Count the set S in two ways:

S={(α,g) in ΩxG : αg=α}.

For corollary, assume not and count the above sum.