Transitive Group Action

Transitive Group Action

Solution 

GROUP THEORY PROBLEM SOLUTION
See defining a permutation representation of it is equivalent to find/define a well-defined GROUP ACTION from .

So Let us first define the Group Action.

We will try to give an example and then develop the action.

Take an element of , say (abc). If you just apply it on an element of X, see what happens(say 0 and 1)

(abc):0->0

(abc):1->9

What I am doing is “ Just apply the Permutation of (abc) on an element of X and see to which element of X it goes)

Formally,

“Apply the permutation on an element of X and See the new partition, which is the image of the action”.

THIS DEFINITION OF THIS ACTION IS THE MAIN PART OF THE SOLUTION OF THE PROBLEMS BELOW.

Assume

FAITHFULLNESS:

See, It is easy to check that no single permutation of   can take every element of X to the same element [So Taking every partition to the same partition is equivalent to the statement]

TRANSITIVITY:

See that from every partition to another partition there is a Permutation of Z involved.

So for all x,y  of X, there exists g of G  such that gx=y.

Note: Take 2 distinct elements from X: say x and y.

See that In simple terms x ->y ,by just a simple transposition on the elements of  X.[() denotes transposition]

0 1 2 3 4 5 6 7 8 9
0 Nothing (cd) (ce) (cf) (bd) (be) (bf) (af) (ae) (ad)

1.9.1

The pointwise stabilizer of abc in is

{a,b,c,Sym(d,e,f)} [where it is the permutation of { a,b,c,d,e,f}.

It is Isomorphic toSo,its order in is 3!=6

  1. The Setwise Stabilizer of abc in is

{Sym(a,b,c),Sym(d,e,f)} [where it is the permutation of { a,b,c,d,e,f}.

So it is 3!x3!=36.

  1. acts transitively on X. So the orbit of is 1 [G.x=X]. So Order of the Stabilizer is equal to the (Order of the group acting)/|X|= 6!/10= 72= .[ By Order-Stabilizer Theorem][|X|| |=|G|]

[Let us list the stabilizer of 0: ={ All elements of G which fixes 0}

If the set  {a,b,c,d,e,f}->{ Permute(a,b,c,),Permute(d,e,f)}/

{Permute(d,e,f),Permute(a,b,c)}]:

So In Total 36+36=72 elements.

1.9.2

Action of 0 1 2 3 4 5 6 7 8 9
Effect of action of (abc) on X (abc) 0 9 8 7 1 2 3 6 5 4
Effect of action of (def) on X (def) 0 2 3 1 5 6 4 9 7 8

To check what happens:    (abc):X->X          (def):X-> X

To Show thatis 2-transtive.

We will show that is 2-transitive under this action.[As they are equivalent as before]

We use the Lemma 1.3.6.

We will show is transitive on X-{0}.

We may compute := the Stabilizer of both 1 and x where x is nonzero and is an element of X.

See that for x=1.

={Permutations such that ab goes to ab(2) and ef to ef (2)/ ab goes to ef(2)  and ef goes to ab(2) and (cd)(1)}