# Equivalent Class

1. Nov 1, 2007

### Magister

Is it possible to know how many equivalent class has an arbitrary group?

2. Nov 1, 2007

Your question is not very clear, can you try to restate it?

3. Nov 1, 2007

### mathwonk

do you mean conjugacy classes?

4. Nov 2, 2007

### Magister

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,

$$Q=\{\pm 1, \pm i, \pm j, \pm k\}$$

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

5. Nov 2, 2007

### matt grime

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'.

6. Nov 2, 2007

### Magister

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?

7. Nov 2, 2007

### matt grime

You still haven't defined "equivalent class".

8. Nov 3, 2007

### Magister

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.