Group actions of subgroup of S_3 onto S_3

[cmj1988]

Given a subgroup of G=S₃={(1)(2)(3), (1 2)(3)} acting on the set S₃ defined as g in G such that gxg^-1 for every x in S₃. Describe the orbit.

The first one is (1)(2)(3)x(3)(2)(1). This orbit is just the identity.

For the second one, I'm not sure how to describe (1 2)(3) except by multiplying it out.

(1 2)(3)(1)(2)(3)(2 1)(3) = (1)(2)(3)
(1 2)(3)(1 2)(2 1)(3) = (1 2)(3)
(1 2)(3)(1 3)(2 1)(3) = (1)(3 2)
(1 2)(3)(2 3)(2 1)(3) = (1 3)(2)
(1 2)(3)(1 2 3)(2 1)(3) = ?
(1 2)(3)(1 3 2)(2 1)(3) = ?

 

[aPhilosopher]

First four look good.

(1 2)(3)(1 2 3)(2 1)(3) = ?
(1 2)(3)(1 3 2)(2 1)(3) = ?

What's the problem? You know how to compose conjugations. These two are exactly like the ones you were able to get.

 

[Dick]

Aside from permutation issues, I don't think you are using the word 'orbit' correctly. If you write your subgroup as G={e,g} (e=identity, g=(12)) then the orbit of an element a of S3 is {eae^(-1),gag^(-1)}. That's a subset of S3 with either one or two elements. That subset is the orbit of a. Once you finish your table you should be able to split S3 into orbits.