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## Monday, November 26, 2007

### Conjugate permutations

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 comment:

1. Hi all,

Firstly thank you all for maintaining this forum. I have actually learnt a lot from it over the last few months, finally decided to join.

Hopefully I wil be able to give back to members as much as I have gained.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)