Jump to content

Permutation group conjugates


Recommended Posts

Hey,

 

I just have a small question regarding the conjugation of permutation groups.

 

Two permutations are conjugates iff they have the same cycle structure.

 

However the conjugation permutation, which i'll call s can be any cycle structure. (s-1 a s = b) where a, b and conjugate permutations by s

 

My question is, how can you find out how many conjugation permutations (s) are within a group which also conjugate a and b.

 

So for example (1 4 2)(3 5) conjugates to (1 2 4)(3 5) under s = (2 4), how could you find the number of alternate s's in the group of permutations with 5 objects?

 

Would it be like

 

(1 4 2) (3 5) is the same as (2 1 4) (35) which gives a different conjugation permutation,

another is

 

(4 1 2)(3 5), then these two with (5 3) instead of ( 3 5),

 

so that gives 6 different arrangements, and similarly (1 2 4) (35) has 6 different arrangements,

 

and each arrangement would produce a different conjugation permutation (s)

 

so altogether there would be 6x6=36 permutations have the property that

s-1 a s = b ?

 

Would each of the arrangements produce a unique conjugation permutation (s) ?

I went through about 6 and I got no overlapping conjugation permutations but I find it a little hard to a imagine there would be unique conjugation permutations for each of the 36 arrangements.

 

Thanks in advance

Link to comment
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now
×
×
  • Create New...

Important Information

We have placed cookies on your device to help make this website better. You can adjust your cookie settings, otherwise we'll assume you're okay to continue.