I know that every finitely generated abelian group can be expressed as a direct sum of cyclic groups. I wondering how easily we can find the cyclic groups given an abelian group. Specifically, one of the form: $$G=\langle x_0,x_1,\cdots, x_n\rangle \big/ \langle f_0,\cdots, f_m\rangle\quad (*)$$ where each $f_i$ is a $\mathbb{Z}-$linear combination of the $\{x_i\}$. The problem I am trying to solve is: $$\langle x,y,z\rangle \big / \langle 6x+10y, 6x+15y,10y+15z\rangle$$ I am not asking for a solution to this particular problem, but I also do not know where to start - and would also be interested in a method for tackling similar problems.

Is there a general strategy for expressing groups of the form $(*)$ as $\oplus C_{a_k}?$ If not, what sort of properties of the $f_i$ can we exploit to do the job?

Thanks for any pointers.


Yes: express the presentation (*) as an integer matrix and compute its Smith normal form.

This is explained in many books. Here are two that I know:

  • Jacobson's Basic Algebra I, chapter 3.

  • Finitely Generated Abelian Groups and Similarity of Matrices over a Field by Christopher Norman.


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