Linear Algebra problem: What does it mean by ZxZxZ are represented by column vectors?

[Steven G]

I am not sure what it means by ZxZxZ are represented by the column vectors. I know which column vectors that is be referred to. I also have no idea what they mean by Z^3/MZ^3. I can't show any work since I don't know the meaning of what is given.
As always, any help would be appreciated.
Steven

 

[topsquark]

Write an element of $\mathbb{Z}^3$ as $\begin{pmatrix} \mathbb{Z} \\ \mathbb{Z} \\ \mathbb{Z} \end{pmatrix}$.

An element $z \in \mathbb{Z}^3 / M \mathbb{Z}^3$ is defined by Mz = 0, similar to $6 \in \mathbb{Z} / 3 \mathbb{Z}$.

-Dan

 

[blamocur]

I think you need to find some generator element $g$ for $\mathbb Z^3/M\mathbb Z^3$. For such element $32g = Mx$ for some $x \in \mathbb Z^3$, but no such $x$ can be found for $ng$ when $n\mod 32 \neq 0$. My quick and dirty script found such element, and since I don't expect an elite member to cheat I put it in a spoiler

Spoiler: Generator element

$g = (4,5,12)$

 

[blamocur]

I think you need to find some generator element $g$ for $\mathbb Z^3/M\mathbb Z^3$. For such element $32g = Mx$ for some $x \in \mathbb Z^3$, but no such $x$ can be found for $ng$ when $n\mod 32 \neq 0$. My quick and dirty script found such element, and since I don't expect an elite member to cheat I put it in a spoiler

Spoiler: Generator element

$g = (4,5,12)$

My script was quick but too dirty, i.e., wrong. Accidentally it did find a correct generator, but half of all elements in $\mathbb Z_{32}$ are generators. I.e., there is a much simpler generator candidate element $g$ for $\mathbb Z/ M \mathbb Z$, i.e., such that $32g \in M \mathbb Z$, but $ng \notin M \mathbb Z$ for $0< n < 32$.

One still has to prove that each element in $\mathbb Z/ M \mathbb Z$ can be represented as $ng$ for some $0 \leq n < 32$. For example, by proving that the group has no more than 32 elements.

 

[blamocur]

Anyone knows how to prove (or disprove?) that the number of elements in $\mathbb Z/M\mathbb Z$ is equal to $\det M$ ?