# Transpositions in algebra

1. Feb 25, 2012

### gotjrgkr

1. The problem statement, all variables and given/known data
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$\geq$2 must be added in the above theorem.
Am I wrong?? If so, could you explain why??

2. Relevant equations

3. The attempt at a solution

2. Feb 25, 2012

### I like Serena

Hi gotjrgkr!

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

3. Feb 25, 2012

### 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?

4. Feb 25, 2012

### Fredrik

Staff Emeritus
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.

5. Feb 25, 2012

Thanks!!!