# Permutation group conjugates

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### #1 Grayham

Grayham

Lepton

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Posted 12 March 2012 - 01:44 PM

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.