Let G be a non-abelian group and let H = ℤ₅. Is the direct product G × H abelian?
AYes — ℤ₅ is abelian, and its commutativity forces the product to be abelian
BIt depends on the specific elements chosen; some pairs commute and some don't
CNo — G × H is abelian only if both G and H are abelian
DYes — direct products always produce abelian groups regardless of the factors
G × H is abelian if and only if both factors are abelian. If G is non-abelian, there exist g₁, g₂ ∈ G with g₁g₂ ≠ g₂g₁. Then (g₁, e_H)(g₂, e_H) = (g₁g₂, e_H) ≠ (g₂g₁, e_H) = (g₂, e_H)(g₁, e_H). The non-commutativity lifts directly to the product. One abelian factor is not enough to rescue the product.
Question 2 Multiple Choice
ℤ₄ × ℤ₂ has order 8. Is it isomorphic to ℤ₈?
AYes — both groups have order 8, so they must be the same group
BYes — the product of cyclic groups is always cyclic
CNo — ℤ₄ × ℤ₂ is not cyclic because gcd(4, 2) ≠ 1
DNo — direct products are never isomorphic to cyclic groups
The Chinese Remainder Theorem for groups states that ℤₘ × ℤₙ ≅ ℤₘₙ if and only if gcd(m, n) = 1. Since gcd(4, 2) = 2 ≠ 1, the isomorphism fails. ℤ₄ × ℤ₂ has no element of order 8 (the maximum order of any element is lcm(4,2) = 4), so it cannot be cyclic. Compare with ℤ₂ × ℤ₃ ≅ ℤ₆, which works because gcd(2, 3) = 1.
Question 3 True / False
ℤ₂ × ℤ₃ is isomorphic to ℤ₆ because gcd(2, 3) = 1.
TTrue
FFalse
Answer: True
True. When the orders of two cyclic groups are coprime, their direct product is again cyclic of order equal to the product. This is the group-theoretic version of the Chinese Remainder Theorem. In ℤ₂ × ℤ₃, the element (1, 1) has order lcm(2, 3) = 6, so it generates the entire group — confirming it is cyclic of order 6, i.e., isomorphic to ℤ₆.
Question 4 True / False
In the direct product G × H, the G-component of a product depends on the H-components of the factors being multiplied.
TTrue
FFalse
Answer: False
False. This is the defining feature of the direct product: the two components operate independently. (g₁, h₁)(g₂, h₂) = (g₁g₂, h₁h₂), where the G-components multiply using G's operation and the H-components multiply using H's operation, with no interaction between them. Neither component 'knows about' the other. This independence is what makes direct products such clean decomposition tools.
Question 5 Short Answer
Explain why the Classification Theorem for Finite Abelian Groups relies on the concept of direct products.
Think about your answer, then reveal below.
Model answer: The Classification Theorem states that every finite abelian group is isomorphic to a direct product of cyclic groups of prime-power order: G ≅ ℤ_{p₁^{a₁}} × ℤ_{p₂^{a₂}} × ⋯ × ℤ_{pₖ^{aₖ}}. The theorem relies on direct products because it decomposes G into independently operating cyclic pieces, each of which is already fully understood. Without the direct product construction, there would be no standard form to classify groups into — the theorem gives a complete, non-redundant list of all finite abelian groups by specifying their prime-power cyclic factors. Recognizing that a group encountered in practice is secretly such a product is the analytical payoff.
The key structural insight is that a finite abelian group can be peeled apart into components that don't interact with each other — the p-primary components for each prime p dividing the group order. Each p-primary component is itself a direct product of cyclic groups of p-power order. The direct product construction is what makes 'independent components' precise.