Counting Equivalent Classes in Arbitrary Groups

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Discussion Overview

The discussion revolves around the concept of equivalent classes and representations in the context of arbitrary groups, specifically focusing on the quaternion group. Participants explore the definitions and implications of equivalent representations and classes, as well as the nature of these representations within the group.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant initially questions the ability to determine the number of equivalent classes in an arbitrary group.
  • Another participant seeks clarification on whether the original question pertains to conjugacy classes.
  • A participant expresses confusion regarding the concept of equivalent representations and whether the quaternion group has a finite number of such representations.
  • It is proposed that there are infinitely many isomorphic representations for any given representation, leading to a distinction between representations and presentations.
  • Further clarification is sought on whether an equivalent class consists of multiple equivalent representations or if a single representation suffices to define each class.
  • One participant suggests that an equivalence class is formed by elements related through similarity transformations, equating this to conjugate classes, and identifies the conjugate elements of the quaternion group.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the definitions of equivalent classes and representations. There is no consensus on the original question, and multiple interpretations of equivalent classes and representations are present.

Contextual Notes

There are unresolved definitions and assumptions regarding what constitutes an equivalent representation and class, as well as the specific nature of the quaternion group's representations.

Magister
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Is it possible to know how many equivalent class has an arbitrary group?
 
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Your question is not very clear, can you try to restate it?
 
do you mean conjugacy classes?
 
Sorry, I was a bit sleepy and very tired when I wrote that post.

What I wanted to ask was, is there a way to know how many equivalent representations a group can have?
For example the quaternions group,

<br /> Q=\{\pm 1, \pm i, \pm j, \pm k\}<br />

Has this group a finite number of equivalent representations?

May be I am not understanding at all what is an equivalent representation and equivalent class...

Thanks for the replies
 
Given any representation there is a proper class of isomorphic representations, never mind finitely many. Do you mean representations or do you mean presentations? In anycase, the answer is 'no, there will be infinitely many (equivalent) presentations'.
 
Just what I though. The problem is that I am asked to find the equivalent classes of the quaternion group and so I am confused. An equivalent class is form of many equivalent representations (and I do mean representation) or I just need one to specify each class?
 
You still haven't defined "equivalent class".
 
I think I have finally got it. An equivalence class is formed of elements which are related to each other by some similarity transformation. This is equivalent to a conjugate class and so my problem then resumes in finding the conjugate elements of the quaternions group which are {1,i,j,k}.

Thanks for you participation.
 

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