Permutations and Transpositions

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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

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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

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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.
 

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