GATE 2023 MA – Question 54
Let G be abelian and Φ:G→(ℤ,+) a surjective homomorphism, with Φ(a)=1. P:for every g∈G there is n∈ℤ with gaⁿ∈ker Φ. Q:G=(ker Φ)⟨a⟩ and (ker Φ)∩⟨a⟩={e}. Select all correct statements.
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Correct answer: (A) P true; (C) Q true
Explanation
Choose n=−Φ(g), proving P. Then g=(ga^(-Φ(g)))a^(Φ(g)) proves the product decomposition. If a^m lies in the kernel then m=Φ(a^m)=0, so the intersection is trivial. Both statements hold. **A,C**.