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 [imath]\mathbb{Z}^3[/imath] as [imath]\begin{pmatrix} \mathbb{Z} \\ \mathbb{Z} \\ \mathbb{Z} \end{pmatrix}[/imath].

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

-Dan

 

[blamocur]

I think you need to find some generator element [imath]g[/imath] for [imath]\mathbb Z^3/M\mathbb Z^3[/imath]. For such element [imath]32g = Mx[/imath] for some [imath]x \in \mathbb Z^3[/imath], but no such [imath]x[/imath] can be found for [imath]ng[/imath] when [imath]n\mod 32 \neq 0[/imath]. 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

[imath]g = (4,5,12)[/imath]

 

[blamocur]

I think you need to find some generator element [imath]g[/imath] for [imath]\mathbb Z^3/M\mathbb Z^3[/imath]. For such element [imath]32g = Mx[/imath] for some [imath]x \in \mathbb Z^3[/imath], but no such [imath]x[/imath] can be found for [imath]ng[/imath] when [imath]n\mod 32 \neq 0[/imath]. 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

[imath]g = (4,5,12)[/imath]

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

One still has to prove that each element in [imath]\mathbb Z/ M \mathbb Z[/imath] can be represented as [imath]ng[/imath] for some [imath]0 \leq n < 32[/imath]. 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 [imath]\mathbb Z/M\mathbb Z[/imath] is equal to [imath]\det M[/imath] ?