General question about orbits (group theory)

In summary, an orbit in group theory is a set of elements related through the group's operations, while a stabilizer is a subgroup that keeps a specific element fixed. Orbits are crucial in understanding the structure and properties of groups and can be used in various applications. The size of an orbit can be calculated using the orbit-stabilizer theorem, and the concept of orbits can be applied to other mathematical structures.
  • #1
linda300
61
3
Hi everyone,

I was just wondering if you have an element of a G-set which has an orbit of only itself,

Say Orbit(e) = {e}, in the set of permutations where G is the same order group of permutations and the operation is conjugation.

This is an Orbit of size 1 correct?

I was just wondering if these size Orbits are often ignored?
 
Mathematics news on Phys.org
  • #2
Can you give an example of when you ever ignore these orbits?
 

FAQ: General question about orbits (group theory)

What is an orbit in group theory?

An orbit in group theory refers to the set of all elements that can be reached from a given element by applying the group's operation. In other words, it is the set of elements that are related to each other through the symmetry operations of the group.

How is an orbit different from a stabilizer?

An orbit and a stabilizer are two important concepts in group theory. An orbit is a set of elements that are related through the group's operations, while a stabilizer is a subgroup that keeps a particular element fixed. In other words, the stabilizer of an element contains all the group's operations that do not change that element.

What is the significance of orbits in group theory?

Orbits are essential in understanding the structure and properties of groups. They help to identify the symmetries present in a group and can be used to classify different types of groups. Additionally, orbits play a crucial role in applications such as crystallography and quantum mechanics.

How do you calculate the size of an orbit?

The size of an orbit can be calculated by using the orbit-stabilizer theorem, which states that the size of an orbit is equal to the index of the stabilizer subgroup. This index can be found by dividing the order of the group by the size of the stabilizer subgroup.

Can the concept of orbits be applied to other mathematical structures?

Yes, the concept of orbits can be extended to other mathematical structures, such as vector spaces, rings, and fields. In these cases, the orbit refers to the set of all elements that can be obtained by applying a transformation to a given element. This concept is widely used in linear algebra and abstract algebra.

Similar threads

I Subgroup axioms for a symmetric group
Replies
1
Views
930
I What are the irreducible representations of point groups and how do I find them?
Replies
6
Views
7K
What Are Symmetric Groups and Their Mathematical Significance?
Replies
1
Views
2K
Group Theory: Definition, Equations, and Examples
Replies
1
Views
2K
I Breaking down SU(N) representation into smaller groups
Replies
3
Views
3K
How Can You Find the Number of Conjugation Permutations in a Group?
Replies
4
Views
3K
From circular orbit to elliptical orbit
Replies
3
Views
2K
I Clarification of a specific orbit example
Replies
3
Views
1K
What is a Subgroup? Definition, Equations & Explanation
Replies
1
Views
2K
What is the Definition and Explanation of a Quotient Group?
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top