- #1
namegoeshere
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Homework Statement
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.
Homework Equations
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.