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Quotient group of S3

  1. Apr 9, 2013 #1
    1. The problem statement, all variables and given/known data
    Let (G,◦) be a group and let N be a normal subgroup of G. Consider the set of all left cosets of N in G and denote it by G/N:

    G/N = {x ◦ N | x ∈ G}.

    Find G/N:

    (G,◦) = (S3,◦) and N = <β> with β(1) = 2, β(2) = 3, β(3) = 1.

    2. Relevant equations



    3. The attempt at a solution

    I'm not sure I understand this problem.

    The permutations are (1)(2)(3), (1,2)(3), (1)(2,3), (1,3,2), (1,3)(2), and (1,2,3).

    Does β(1) = 2, β(2) = 3, β(3) = 1 mean N is the permutation (1,2,3)? Can't I compose (1,2,3) with any x ∈ G, and receive x ◦ N ∈ G with the resulting quotient group being {x | x ∈ G}, since N is onto and one-to-one?

    Sorry if that makes no sense. I'm confused, to say the least.
     
  2. jcsd
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