The Symmetric Group S₃

Abstract Algebra · Axiom Academy

EXAMPLE The Symmetric Group S₃ Explore all permutations of three objects, construct the Cayley table, and discover connections to triangle symmetries. S₃ has several proper subgroups. Click all sets that form subgroups under composition: Identity and one transposition Excellent work! You've systematically constructed S₃. Here's what we discovered: S₃ has 6 elements: |S₃| = 3! = 6 permutations of 1, 2, 3 Two types of elements: 3 transpositions (2-cycles) and 2 three-cycles, plus the identity Cycle notation is efficient: (123) is more compact than writing out the full mapping Non-abelian structure: Composition order matters: (12)(23) ≠ (23)(12) Subgroup structure: S₃ has 6 subgroups total: e , three order-2 subgroups, A₃, and S₃ itself Geometric connection: S₃ is isomorphic to the dihedral group D₃ (symmetries of a triangle) S₃ is the smallest non-abelian group! It serves as a fundamental example in abstract algebra, demonstrating how permutations form groups and connecting algebraic structures to geometric symmetries.

This is the written version of the interactive lesson above. See the full Abstract Algebra course.