equivalence of submodules

Question

I have Z^3/M = Z^3/N = Z_k where M,N are submodules of Z^3 and Z_k is cyclic order k.

I would like to say some SL_3(Z) transformation takes M to N. Is this true? How to show?

Answer 1

It is enough to show that

if $M\subseteq \mathbb Z^3$ be a subgroup such that $\mathbb Z^3/M$ is a cyclic group of order $k$, then there exists $g\in\mathrm{SL}(3,\mathbb Z)$ such that $g(M)=\langle (1,0,0),(0,1,0),(0,0,k)\rangle$.

Let $M\subseteq \mathbb Z^3$ be a subgroup such that $\mathbb Z^3/M$ is a cyclic group of order $k$. Then $M$ is free of rank $3$, and there exists $A\in M(3,\mathbb Z)$ such that $M=A\cdot\mathbb Z^3$. Using the Smith normal form, we know that there exists $3\times 3$ matrices $P$ and $Q$, invertible over $\mathbb Z$, such that $PAQ=D$ with $D=\left(\begin{smallmatrix}a\\\&b\\\&&c\end{smallmatrix}\right)$ and $a\mid b\mid c$. Then $PM=PAQ\mathbb Z^3=D\mathbb Z^3$.

It follows that $P\in\mathrm{SL}(3,\mathbb Z)$ is such that $PM$ is generated by $(a,0,0)$, $(0,b,0)$ and $(0,0,c)$ with $a\mid b\mid c$. Since $\mathbb Z^3/g(M)$ is cyclic of order $k$, we must have $a=b=1$ and $c=k$. This tells us that the claim above is true.

(I've done everything at the level of generality which your problem needs, and I'll leave the fun of finding the correct general statement for you...)