Groups of Order 6

Abstract Algebra · Axiom Academy

Complete classification theorem: there are exactly two groups of order 6 up to isomorphism—the cyclic group Z₆ and the symmetric group S₃. 1. The Classification Question Our strategy will be to analyze the possible structures by examining element orders and using Lagrange's theorem. We'll see that a group of order 6 must be either cyclic or closely related to permutations. Before diving into the classification, we preview a powerful result from Sylow theory that guarantees the existence of certain subgroups: A subgroup of order 2 (Sylow 2-subgroup) A subgroup of order 3 (Sylow 3-subgroup) This means every group of order 6 contains elements of orders 2 and 3. The question becomes: how are these elements related? 3. Case Analysis: Cyclic vs Non-Cyclic We divide the analysis into two exhaustive cases based on whether the group has an element of order 6: If: G has an element g of order 6 If: G has no element of order 6 Then: All non-identity elements have order 2 or 3 The symmetric group on 3 elements. Why does the non-cyclic case yield S₃? Let's examine the structure more closely. If G has no element of order 6, then by Cauchy's theorem it has elements a, b with: The six elements are: . The multiplication rules are determined by how a and b interact. A careful analysis shows this structure is isomorphic to S₃, the group of permutations of three objects. 3 transpositions (swaps): order 2 5. Complete Classification Theorem Z₆: The cyclic group (abelian)

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