Permutations and Transpositions

[wubie]

Hello,

I am a little confused about an example. By definition,

A cycle of m symbols CAN be written as a product of m - 1 transpositions.

(x1 x2 x3 ... xn) = (x1 x2)(x1 x3)...(x1 xn)


Now

Express the permutation (23) on S = {1,2,3,4,5} as a product of transpositions.


(23) = (12) o (23) o (13) = (12) o (13) o (12)


I can see how it works. But based on the def. I don't see how they came up with the answer. I know this is simple but I don't see it. What the hey?

 

[Hurkyl]

I'm a little confused; (2, 3) is a product of transpositions...

can you provide a little more of the context?

 

[wubie]

I am confused too.

This is in Schaum's Outlines of Modern Abstract Algebra. It is in Chapter 2: Relations and Operations, under the section Permutations.

The question/ example above is exactly as it is in the book.
I know that a permutation can be expressed as a product of transpositions. And that there can be more than one way to express a permutation as a product of transpositions. I think that is what they are trying to show.

However I don't understand the method in which they selected these particular transpositions to express the permutation (23). I can see that it works out. But why/how did they know that (23) was a product of the above transpositions? Trial and error?

 

[Hurkyl]

Ah, an example; it makes more sense now.

Anyways, I can see an algorithmic procedure that gives you the second example, but I'm tongue-tied trying to explain it... if you limit yourself to the condition that each transposition must have '1' in it, you could probably figure the procedure out for yourself.


I can motivate the first one from products of transpositions:
(12) (23) = (23) (13)
so
(23) = (12) (23) (13)

then again, they might simply just be examples without expecting any motivation.

 

[wubie]

It just confused me since the way they got the product of transpositions for (23) wasn't based on the defintion.

(x1 x2 x3 ... xn) = (x1 x2)(x1 x3)...(x1 xn)

I mean, using the def. I couldn't see how one could come up with

(23) = (12) o (23) o (13) = (12) o (13) o (12).

Thanks Hurkyl.