Math 5125 Friday, November 18 November 18, Ungraded Homework Exercise 10.5.1 on page 403 Suppose that A ψ---→ B φ---→ C α y β y γ y A0 ψ0---→ B0 φ0---→ C0 is a commutative diagram of groups and that the rows are exact. Prove that (a) if φ , α are surjective and β is injective, then γ is injective. (b) if ψ0 , α and γ are injective, then β is injective, (c) if φ , α and γ are surjective, then β is surjective, (d) if β is injective and α , φ are surjective, then γ is injective, (e) if β is surjective and γ , ψ0 are injective, then α is surjective. (a) Let c ∈ ker γ . Then c = φ b for some b ∈ B and 0 = γ c = γφ b = φ0 β b , so β b ∈ ker φ0 . Therefore β b = ψ0 a0 for some a0 ∈ A0 , and then we may write a0 = α a for some a ∈ A because we are given that α is surjective. Therefore βψ a = ψ0 α a = ψ0 a0 = β b , consequently ψ a = b , because we are given that β is injective. We conclude that c = φ b = φψ a = 0 as required.
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