Group actions of subgroup of S_3 onto S_3

In summary, the given subgroup of G=S3 acts on the set S3 by conjugation, where the orbit of an element is the subset of S3 obtained by applying the subgroup's elements to the element. The orbit of the first element, (1)(2)(3), is simply the identity element. The orbit of the second element, (1 2)(3), can be described by multiplying it out and obtaining the elements (1 2)(3) and (1 3 2). The remaining elements in S3 are either in their own individual orbits or in the same orbit as one of these two elements.
  • #1
cmj1988
23
0
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) = ?
 
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  • #2
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.
 
  • #3
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.
 

1. What is a group action?

A group action is a mathematical concept that describes how a group (a set of objects and a defined operation) acts on another set of objects. This action can be thought of as a way of moving or transforming the objects in the second set according to the operation defined by the group.

2. What is S_3?

S_3 is the notation for the symmetric group of order 3, which is the group of all possible permutations (rearrangements) of three objects. It has a total of 6 elements: the identity permutation and 5 other permutations.

3. How is a subgroup of S_3 defined?

A subgroup of S_3 is a subset of the elements of S_3 that also forms a group under the same operation. In other words, it is a smaller group within S_3 that still follows the same rules and properties as S_3.

4. How does a subgroup of S_3 act on S_3?

A subgroup of S_3 can act on S_3 by performing permutations on the elements of S_3. This means that each element in the subgroup will correspond to a specific permutation of the elements in S_3, and this permutation will be applied to the elements in S_3 when the subgroup acts on it.

5. What is the significance of group actions of subgroup of S_3 onto S_3?

The group actions of subgroup of S_3 onto S_3 have important applications in various areas of mathematics, including group theory, abstract algebra, and combinatorics. They can also be used to study and classify different groups and their properties.

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