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    Every finitely generated torsion-free abelian group is free abelian

    P-QQTBQ

    Area
    Algebra
    Topics
    Abelian Groups, Free Modules, Torsion
    Status
    draft

    Prove that every finitely generated torsion-free abelian group is free abelian.

    Solution

    By the structure theorem for finitely generated abelian groups, \[ G\cong \mathbb Z^r\oplus \mathbb Z/d_1\mathbb Z\oplus\cdots\oplus\mathbb Z/d_s\mathbb Z \] with \(d_i\ge2\). The finite direct summand is exactly the torsion subgroup of \(G\). If \(G\) is torsion-free, then \(s=0\), so \[ G\cong\mathbb Z^r. \] Hence every finitely generated torsion-free abelian group is free abelian.

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