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Abstract Algebra Proof question

  1. Nov 16, 2008 #1
    1. The problem statement, all variables and given/known data
    The question is:
    Let A be a subset of Sn that contains all permutations alpha such that alpha can be written as a product of an even number of transpositions. Prove that A is a group with product of permutations.

    I understand what I need to do to prove it, but I am not sure how to start it. Do I use:
    Let alpha=(a1a2a3...an) and
    beta=(b1b2b3...bn) and try to find closure,

    alphabeta=(a1a2a3...an)(b1b2b3...bn)=(a1a2)(a1a3)...(a1an-1)(a1an)(b1b2)(b1b3)...(b1bn-1)(b1bn),

    or am I going about it the wrong way?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 16, 2008 #2

    morphism

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    What is a group? There are some axioms A must satisfy in order for it to be called one. For instance, the product of two things in A must also lie in A. Translating this to the situation at hand: if two permutations can be written as a product of an even number of transpositions, can their product be as well? Obviously yes! Now check the other axioms similarly.
     
  4. Nov 17, 2008 #3

    HallsofIvy

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    Where did you use the fact that they can be written as an even number of transpositions?
     
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