Finding conjugacy classes

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In summary, the conversation discusses finding conjugacy classes and the most efficient method for doing so. The speaker suggests writing out the orbit of x and using the set relation yx=(x^2)y to work out each conjugate individually. They also mention the use of a theorem that can save time in certain cases. The conversation ends with a recommendation to use GAP, a tool that can compute conjugacy classes.
  • #1
catcherintherye
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I was wondering if anyone knows a more efficient method of finding conjugacy classes than the one i am currently using.
tex/ Example D_6* =<x,y| x^3=1, y^4=1, yx=x^2y>

now to find the conjugacy classes of this group i would first write out
the orbit of x <x> ={ 1x1, xxx^2, x^2xx, yxy^3,...x^2y^2xy^2x^2,...etc}

then i would use the set relation yx=(x^2)y to work out each of these 12 conjugates individually. Once this is done i continue with <x^2>, <y> etc...

..the only short cut i have found is the theorem that says <x>intersection<y> = empty set or <x>=<y>. But even with this surely there is a quicker way??
 
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  • #2
Hello,

In terms of finding the conjugacy classes, there doesn't look to be a faster way. I mean, if you actually wanted to know what they are, how would you get them without doing the calculations? If you don't want to do them by hand, I recommend GAP: http://www-gap.mcs.st-and.ac.uk/
It will compute conjugacy classes for you.

There are some other theorems that might save you some time. For example, if G has odd order g, and if h is the number of conjugacy classes of G, then g = h (mod 16). Once you have computed most of the classes, this will probably tell you if the rest of the elements form a single conjugacy class or not. There is another variant of this theorem which says that if all of the primes dividing g are congruent to 1 (mod m), then g = h (mod 2m^2).
http://links.jstor.org/sici?sici=0002-9890(199505)102%3A5<440%3ACRTOOA>2.0.CO%3B2-%23
 
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  • #3


There are a few different methods for finding conjugacy classes in a group, and the most efficient one will depend on the specific group and its defining elements. However, here are a few general strategies that may help in finding conjugacy classes more efficiently:

1. Use known properties and theorems: As you mentioned, the theorem stating that the intersection of two subgroups is either the identity element or the larger subgroup can help to narrow down the number of conjugacy classes to consider. Additionally, if you know any other properties or theorems that apply to your specific group, you can use them to simplify the process.

2. Look for patterns: In some cases, you may be able to identify patterns within the conjugacy classes that can help you to group them together. For example, in the group you provided, the elements x and x^2 have the same orbit, so you can group them together as a single conjugacy class.

3. Use group generators: If you have a set of generators for your group, you can use them to generate the entire group and then look for conjugacy classes within that generated group. This can be a more efficient approach when working with larger or more complex groups.

4. Utilize software or online resources: There are many online resources and software programs that can help to find conjugacy classes in a given group. These tools can often handle more complex groups and may be able to find conjugacy classes more quickly than manual methods.

Overall, the most efficient method for finding conjugacy classes will depend on the specific group and its defining elements. It may be helpful to try out a few different approaches and see which one works best for your particular case.
 

1. What is a conjugacy class?

A conjugacy class is a group of elements in a mathematical group that are considered equivalent under the operation of conjugation. This means that two elements are in the same conjugacy class if they can be transformed into each other by multiplying on the left and right by elements from the group.

2. Why is it important to find conjugacy classes?

Finding conjugacy classes allows us to better understand the structure of a mathematical group. It also helps us to determine the size and order of a group, as well as its subgroups and normal subgroups.

3. How do you find conjugacy classes?

To find conjugacy classes, you need to determine the elements that are conjugate to each other. This can be done by finding the centralizer of each element, which is the set of elements that commute with the given element. The centralizer will then contain all the elements that are conjugate to the given element.

4. Can two elements be in multiple conjugacy classes?

No, two elements in a mathematical group cannot be in multiple conjugacy classes. This is because the elements in a conjugacy class are considered equivalent, and an element cannot be equivalent to itself in multiple ways.

5. How does the concept of conjugacy classes apply to real-world situations?

The concept of conjugacy classes can be applied to various fields, such as chemistry and physics. In chemistry, for example, the concept of molecular symmetry is based on the idea of conjugacy classes. In physics, the concept of gauge symmetry is also related to conjugacy classes. Additionally, conjugacy classes can be used in data analysis and machine learning to identify patterns and relationships between data points.

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