Computing S₃/A₃

Abstract Algebra · Axiom Academy

Constructing the quotient group and verifying the isomorphism with Z₂ Excellent work! You've constructed the quotient group S₃/A₃. Here's what we learned: Normal Subgroups Create Quotients: Since A₃ is normal in S₃, we can form the quotient group S₃/A₃ Cosets Partition the Group: The two cosets are the even permutations (A₃) and odd permutations ((12)A₃) Index Determines Size: |S₃/A₃| = |S₃|/|A₃| = 6/3 = 2, giving us a group of order 2 Isomorphism with Z₂: S₃/A₃ ≅ Z₂ because both are cyclic groups of order 2 Quotients "Forget" Information: The quotient remembers only parity (even/odd), not which specific permutation Visual Interpretation: All elements of A₃ collapse to the identity, showing how quotients identify elements This quotient construction is fundamental in group theory. It shows how we can create simpler groups by "modding out" by normal subgroups, similar to how Z/nZ works with integers. The parity homomorphism from S₃ to Z₂ has kernel A₃, making this quotient particularly natural!

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