External Direct Products

Abstract Algebra · Axiom Academy

LESSON External Direct Products Building new groups from existing ones: the construction and properties of G × H with componentwise operations For (g₁, h₁), (g₂, h₂) ∈ G × H, we define: Each component operates independently using its original group operation. The identity element is (e_G, e_H), and the inverse of (g, h) is (g⁻¹, h⁻¹). The direct product has elegant order properties that relate the original groups to the product: Why LCM? For (g, h) to return to (e_G, e_H), we need g^n = e_G and h^n = e_H. The smallest such n is the LCM of their individual orders. The direct product contains natural copies of its constituent groups as subgroups: These subgroups are normal in G × H They intersect trivially: (G × e_H ) ∩ ( e_G × H) = (e_G, e_H) Every element can be written uniquely as their product Let's examine ℤ₄ × ℤ₆ to see these properties in action: (1, 1): order = lcm(4, 6) = 12 (3, 5): order = lcm(4, 6) = 12 How can we tell if a group K is (isomorphic to) a direct product? The recognition theorem gives us the criteria: K has normal subgroups N₁ ≅ G and N₂ ≅ H N₁ ∩ N₂ = e (they intersect trivially) Every element k ∈ K can be written as k = n₁n₂ with n₁ ∈ N₁, n₂ ∈ N₂ Why it works: The theorem ensures that K behaves exactly like a direct product— elements factor uniquely, components commute (because both subgroups are normal), and the structure matches G × H perfectly.

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