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Transpositions in algebra

  1. Feb 25, 2012 #1
    1. The problem statement, all variables and given/known data
    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

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    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


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    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
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