Exploring the Connection Between Right Cosets and Orbits

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In summary, there is indeed a relationship between right cosets and orbits of x. This is evident in the similarities between the properties of cosets and orbits. Additionally, the study of group actions can be divided into two problems: understanding the action within single orbits and understanding how the orbits make up the set X. This is further discussed in theorem 3 on page 4 of an article on group actions.
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cmj1988
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I'm just wondering if there is some sort of relationship between right coset and orbit of x. We just got to cosets, and it seems like the properties of cosets are eerily similar to orbits.
 
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Of course their is a relationship.

First of all, I think you answered your own question.

Second, a random book I picked up had a section on cosets right after the section on orbits and cyclic groups.

Q.E.D.
 
  • #3
I'd like to note, you specifically mentioned the right coset and orbit of 'x'. Be careful if you choose a different variable, say, 'z'.

Z has mystical properties that defy abstract algebra.
 
  • #4
cmj1988 said:
I'm just wondering if there is some sort of relationship between right coset and orbit of x. We just got to cosets, and it seems like the properties of cosets are eerily similar to orbits.
You might be interested in theorem 3 on page 4 of this article.
The study of group actions thus divides neatly into two problems: the internal problem of understanding the action within single orbits (equivalent to studying the canonical action in coset spaces) and the external problem of understanding how the orbits are put together to form the set X.
 

1. What is a right coset?

A right coset is a subset of a group formed by multiplying each element of the subgroup by a specific element of the group. This results in a set of elements that are all equivalent to each other, but may be in a different order.

2. How is a right coset different from a left coset?

A right coset is formed by multiplying the elements of the subgroup on the right, while a left coset is formed by multiplying the elements on the left. This can result in different elements being included in the coset, but the overall structure and properties remain the same.

3. What is the significance of cosets in group theory?

Cosets allow us to partition a group into smaller, equivalent subsets. This can help us to understand the structure and properties of a group, as well as identify relationships between elements and subgroups.

4. How are orbits related to right cosets?

Orbits are a special type of right coset, where the group is acting on itself through a specific operation. The orbit contains all elements that can be reached from a given element in the group by applying the operation. This is similar to how a right coset contains all elements that can be obtained by multiplying the subgroup by a specific element.

5. Can right cosets be used to find subgroups of a group?

Yes, right cosets can be used to identify subgroups of a group. If a subgroup is formed by multiplying a specific element of the group by all elements of the subgroup, then the right coset of that element will be equal to the subgroup. This can help us to identify and understand the structure of a group and its subgroups.

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