Thread: Orbits and Groups

[tex] g=(1 2) (3 4 5 6)
Let G= [tex] {1, g, g^2, g^3}

Compute
Computer both stabilizer and orbit of 1,3,7 in G.

Note:
|stabilizer (x)||orbit (x)|=|G|
Stabilizers is the Action

Originally Posted by Linnus

[tex] g=(1 2) (3 4 5 6)
Let G= [tex] {1, g, g^2, g^3}

Compute
Computer both stabilizer and orbit of 1,3,7 in G.

Note:
|stabilizer (x)||orbit (x)|=|G|
Stabilizers is the Action

The orbit of an element in the set the group is acting upon, S, is everywhere in S it is sent to by an element of the group. So, for example, there is no way of sending 1 to anything other than 2. Similary, g sends 1 to 2.

The stabiliser of an element is the set of all group elements which "stabilise" the element; group elements which don't send the element anywhere.

So, for example, but . Therefore, stabilises 1, as does the group identity. Therefore, the orbit of the element 1 is {1, 2} and its stabiliser is the subgroup . Note that 2.2=4=|G|.

Does that make sense?

Thanks for the help. I understand that part, but I don't know how to compute the orbit and the stabilizer of 3. Any help is appreciated!

[quote=Linnus;505474]
[tex] g=(1 2) (3 4 5 6)
Let G= [tex] {1, g, g^2, g^3}

Compute
g^2 = (1 2) (3 4 5 6)(1 2) (3 4 5 6) = (3 5)(4 6)
g^3 = (3 5)(4 6)(1 2) (3 4 5 6) = (3 6 5 4)

stab(1) = {g^2,g^3}, orb(1) = 2
stab(3) = {1}, orb(3) = 4
stab(7) = {1,g,g^2,g^3}, orb(7) = 1

i am not sure, but thats what i would do

Originally Posted by Linnus

Thanks for the help. I understand that part, but I don't know how to compute the orbit and the stabilizer of 3. Any help is appreciated!

stabilizer of x = consist of those elements of G that send x itself,
which is neutral element in this case (for 3)

Originally Posted by Linnus

Thanks for the help. I understand that part, but I don't know how to compute the orbit and the stabilizer of 3. Any help is appreciated!

I you understand my post, then surely you can apply it to working out the orbit and stabiliser of 3?

What is it, specifically, that you do not understand?