<s>[INST] Linear algebra about cyclic group and torsion factors Express the commutative group $Z^{3}/(f_{1}.f_{2},f_{3})$ as a direct sum of cyclic group where 
$f_{1}=(4,6,9), f_{2}=(2,4,12), f_{3}=(4,8,16)$
my answer is $Z[x]/(-11x^{2}+22x-128)$.
I wonder if my calculation is wrong, for this is not a direct sum of cyclic group, hope someone can give me a more clear answer.
How to find the number of non-isomorphic commutative group of a certain order, like $100$ and $p^{3}$ where $p$ is a prime. 
I know that I can use Kronecker Decomposition Theorem, finite abelian group theorem.
 [/INST] The smith normal form for 
$$\begin{pmatrix} 4 & 6 &9\\ 2 & 4 & 12 \\ 4 & 8 & 16\end{pmatrix}$$ can be found by several steps. We first have by substracting two times middle row from top and bottom row:
$$\begin{pmatrix} 0 & -2 &-15\\ 2 & 4 & 12 \\ 0 & 0 & -8\end{pmatrix}$$ which is the same as 
$$\begin{pmatrix} 2 & 4 & 12 \\0 & -2 &-15\\ 0 & 0 & -8\end{pmatrix}$$
Then we substracting the first column from second and third column, and do the same for second column:
$$\begin{pmatrix} 2 & 0 & 0 \\0 & -2 &-1\\ 0 & 0 & -8\end{pmatrix}$$
and then we have:
$$\begin{pmatrix} 2 & 0 & 0 \\0 & 0 &-1\\ 0 & 16 & -8\end{pmatrix}$$
changing order we have:
$$\begin{pmatrix} 2 & 0 & 0 \\0 & 16 &0\\ 0 & 0 & -1\end{pmatrix}$$
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