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    Subgroups of $\ZZ^3$ and the structure of the quotient

    P-QMMKE

    Area
    Algebra
    Topics
    Abelian Groups, Free Modules, Structure Theorem
    Status
    draft

    Describe the subgroups of \(\ZZ^3\) and the possible structures of the quotient by such a subgroup.

    Solution

    Every subgroup \(H\le\ZZ^3\) is a free abelian group of rank \[ r\in\{0,1,2,3\}. \]

    By Smith normal form, there is a basis \(e_1,e_2,e_3\) of \(\ZZ^3\) and positive integers \[ d_1\mid d_2\mid\cdots\mid d_r \] such that \[ H=d_1\ZZ e_1\oplus\cdots\oplus d_r\ZZ e_r. \] Therefore \[ \ZZ^3/H \cong \ZZ^{3-r} \oplus \ZZ/d_1\ZZ \oplus\cdots\oplus \ZZ/d_r\ZZ. \]

    Thus the structure theorem for finitely generated modules over the PID \(\ZZ\) completely describes both the subgroup, up to change of basis, and the quotient.

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