**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

- 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.

- 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)}