1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Transpositions in algebra

  1. Feb 25, 2012 #1
    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[itex]_{n}[/itex] 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[itex]\geq[/itex]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. jcsd
  3. Feb 25, 2012 #2

    I like Serena

    User Avatar
    Homework Helper

    Hi gotjrgkr! :smile:

    It's called an empty product.
    See: http://en.wikipedia.org/wiki/Empty_product
     
  4. Feb 25, 2012 #3
    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?
     
  5. Feb 25, 2012 #4

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  6. Feb 25, 2012 #5
    Thanks!!!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook