Two permutations are conjugate IFF they have the same cycle structure.

So if we calculate the conjugate of

a=(1 2 3)(4 5) and

b=(3 5),

then we know that the conjugate has the same cycle structure as a. Let's find out:

a^b=(3 5)((1 2 3)(4 5))(3 5)

1 2 3 4 5

1 2 5 4 3 : (3 5) applied

2 5 1 3 4 : (1 2 3)(4 5) applied

2 5 4 3 1 : (3 5) applied

In cycles: (1 2 5)(3 4)

There is a smarter way to get at this result. Permute the cycle structure of a with b:

(1 2 3)(4 5)

(1 2 5)(4 3) = (1 2 5)(3 4).

Using this method we can simply find the conjugating permutation of a and b.

For example

a=(1 2 3)(4 5 6) and

b=(1 3 5)(2 4 6).

Since a and b have the same cycle structure there must be a permutation c such that

a=c b c^-1.

1 2 3 4 5 6 : cycle a as permutation

1 3 5 2 4 6 : cycle b as permutation

We get from a to b by applying cycle

(2 3 5 4) which must be c.

Let's verify.

1 2 3 4 5 6

1 3 5 2 4 6 : apply c

3 5 1 4 6 2 : apply b

3 4 5 6 1 2: apply c^-1 = (2 4 5 3)

In cycles:(1 3 5)(2 4 6) = b.

1-2017 More on the randomness of randomness.

2 months ago