Let $G$ be a finitely generated abelian group with a minimal generating set $S$. Is it possible to find a partition of $S$ to $\{A,B\}$ such that $A$ generates a free abelian subgroup, and $ B $ generates a finite abelian subgroup?
2026-03-26 16:26:50.1774542410
Partition of minimal generating set
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Let $G=\langle x\rangle\times\langle y\rangle\cong C_\infty\times C_2$ and take $S=\{x,xy\}$. There is no partition $S=A\cup B$ with the desired properties.