Chapter 7 - Sub Orbits And Multiple Transitivity Flashcards
If G acts on Ω transitively, consider the action of Gα, the set of g which fixes α, on Ω.
Define a sub orbit and its subdegree. What is the rank of the action?
A suborbit is an orbit of Gα οn Ω.
The size of this is a subdegree
The rank of the action is the number of suborbits.
Note this is always at least 2.
Prove that the rank of an action of G is:
(1/|G|)•Σ|fix(g)|^2
Apply earlier lemma on action of Gα to show the rank is:
(1/|Gα|)•Σ|fix(g)| where we are only summing over Gα.
Now use the fact that: |Gα|=|G|/|Ω|.
Define the 2-point stabiliser of a, b.
Define the point wise stabiliser
Define the set wise stabiliser of a subset Γ<Ω
This is: Ga,b=Ga^Gb.
The point wise stabiliser of a whole collection of elements is just this intersection repeated.
The set wise stabiliser, denoted GΓ, is the subgroup of g which fixes Γ.
Prove that the action of G on Ω is of rank 2 IFF ΩxΩ\Δ is a single orbit, where Δ is the diagonal.
If action is of rank two, then take any two points in ΩxΩ\Δ, and note that we can then how that some element maps these two points to each other.
Conversely, if the action is of rank larger than two, we can show this isn’t the case.
Define what it means for an action of G to be t-transitive on Ω, where t is less than the size of Ω
Prove that G on Ω is t-transitive IFF Gα on Ω{α} is (t-1)-transitive.
Prove the corollary that if G acts t-transitively on Ω (of size n) then |G| is divisible by n,n-1,…,n-t+1
Let Ωt’ be the set of all t-tuples of elements in Ω with all elements in any tuple pointwise different. Then the action of G is t-transitive on Ω if G acts transitively on this set.
Follow through definitions in obvious way.
For what t is Sn t-transitive?
And An?
Sn is obviously n-transitive, and no higher.
We show by induction that An is (n-2) transitive and no higher, for n>2
A3 is generated by a three cycle, so is only 1-transitive. Then An has point stabiliser An-1. Then use our formula for transitivity of point stabilisers and induction.
