Permutation group conjugates

[physicsjock]

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 ?


Thanks in advance

 

[lippyka]

for any help. Yes, that is correct. To find the number of conjugation permutations (s) that are within a group which also conjugate a and b, you would need to look at the number of different arrangements of each of the permutations. As you have demonstrated, for each of the given permutations, there are 6 different arrangements, so altogether there would be 6x6=36 permutations with the property that s-1 a s = b.