Groups, Rings and Modules | Part IB, 2004

State the structure theorem for finitely generated abelian groups. Prove that a finitely generated abelian group AA is finite if and only if there exists a prime pp such that A/pA=0A / p A=0.

Show that there exist abelian groups A0A \neq 0 such that A/pA=0A / p A=0 for all primes pp. Prove directly that your example of such an AA is not finitely generated.

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