General structure quotient of two rank-2 abelian groups

Question

Bernstein, Sloane $\&$ Wright, Discrete Mathematics 170 (1997) 23-29 states the following property concerning a two dimensional lattice $\Lambda$:

Let $\Gamma$ be a sublattice of $\Lambda$ of index $N$ $\ldots$ Since $\Lambda$ has rank 2, the quotient group $Q=\Lambda/\Gamma$ is either a cyclic group $C_N$, $\ldots$, or a direct product $C_{N/m} \times C_{m}$ of a pair of groups with $m$ dividing $N/m$, so $m^2|N$.

Lacking any suitable reference, how does one prove this?

Answer 1

References are legion. It follows, for example, from this (Theorem II.1.6, from Thomas Hungerford's Algebra, Springer-Verlag GTM 73):

Theorem. If $F$ is a free abelian group of finite rank $n$ and $G$ is a nonzero subgroup of $F$, then there exists a basis $\{x_1,\ldots,x_n\}$ of $F$, an integer $r$, $1\leq r\leq n$, and positive integers $d_1,\ldots,d_r$ such that $d_1|d_2|\cdots|d_r$ and $G$ is free abelian with basis $\{d_1x_1,\ldots,d_rx_r\}$.

From this it follows that $F/G$ is isomorphic to $\mathbb{Z}^{n-r}\times C_{d_1}\times\cdots\times C_{d_r}$. If $G$ has finite index, then you must have $n=r$. Your case is the case $n=2$. The quotient will have the form $C_{d_1}\times C_{d_2}$ with $d_1|d_2$, and $d_1d_2=N$. You can then express $d_2=N/d_1$ with $d_1$ dividing $N/d_1$.

Hungerford includes the proof, so you can find it there. Similar theorems can be found in Rotman's An Introduction to the Theory of Groups 4th Edition, Springer-Verlag GTM 148, Theorem 10.21.