Transpositions in Abstract Algebra

[gotjrgkr]

Homework Statement

Hi!
There's a theorem 7.43 in p.221(Hungerford's abstract algebra) which states that every permutation in S_{n} is a product of transpositions.
What I know about the concept of transposition is it is defined if there are at least two distinct elements. But, in the above theorem, there's no assumption to prevent that n could be 1. I think, in that case, saying a product of transpositions is meaningless.
So, I think an assumption such as n\geq2 must be added in the above theorem.
Am I wrong?? If so, could you explain why??

Homework Equations

The Attempt at a Solution

 

[I like Serena]

Homework Statement

Hi!
There's a theorem 7.43 in p.221(Hungerford's abstract algebra) which states that every permutation in S_{n} is a product of transpositions.
What I know about the concept of transposition is it is defined if there are at least two distinct elements. But, in the above theorem, there's no assumption to prevent that n could be 1. I think, in that case, saying a product of transpositions is meaningless.
So, I think an assumption such as n\geq2 must be added in the above theorem.
Am I wrong?? If so, could you explain why??

Homework Equations

The Attempt at a Solution

Hi gotjrgkr!

It's called an empty product.
See: http://en.wikipedia.org/wiki/Empty_product

 

[gotjrgkr]

So, do you mean the theorem also makes sense even when n=1?
Are you sure?? Where can you find this? I mean, do you have a book explaing about it?

 

[Fredrik]

It's just a matter of choosing a definition of "product" that ensures that we don't have to state special cases separately. It's just a convenience. The statement makes sense for n=1 if we want it to.

 

[gotjrgkr]

Thanks!