I need to characterize every finitely generated abelian group G that has the following property: $$\frac{G}{S} \text{ is cyclic for every } S\leq G$$ I know I am supposed to use the structure theorem to reach contradictions about the underlying structure of the decomposition (for example only one or two primes in its decomposition and such). However, I can't seem to figure out precisely how to reach such contradictions with this property.
2026-03-30 03:01:50.1774839710
F.G. abelian group so that every quotient is cyclic
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Taking @Troposphere's hint, we get that $G $ is cyclic. (So it's singly generated.)