Abelian free group $G, H$ have rank $2$ and $G$ has $\Bbb Z$-basis $x, y$, if $H$ has $\Bbb Z$-basis $$2x+y, 2x-3y$$ then what are the elements of $G/ H$ ?
I am new to the topics, so don't know how to start. The question is motivated from the following excerpt of the book Algebraic-Number Theory by Ian Stewart and David Tall, on page 30 -
for example, if $G$ has rank $3$ and $\Bbb Z$-basis $x, y, z$; and if $H$ has $\Bbb Z$-basis $$3x+y-2z, 4x-5y+ z, x +7z,$$ then $|G/ H|$ is the absolute value of $\begin{bmatrix} 3 & 1 & -2\\ 4 & -5 & 1\\ 1 & 0 & 7 \end{bmatrix}$, namely 142.
I wanted to know what are the elements of quotient group $G/H$, and asked it but since it has a large order i have modified to my present question.
I assume that $G$ is a free Abelian group over a two-element set $\{x,y\}$ and $H$ is the subgroup of $G$ generated by $2x+y$ and $2x-3y$. It is easy to check that $H$ has rank $2$. The theorem on subgroups of a finitely generated free Abelian group (see, for instance, [§20, Kur]) implies that there exist bases $\{u_1,u_2\}$ and $\{v_1,v_2\}$ of the groups $G$ and $H$, such that $v_1=k_1u_1$ and $v_2=k_2u_2$ for some natural numbers $k_1|k_2$. It follows that $G/H$ is isomorphic a direct product of cyclic groups of orders $k_1$ and $k_2$.
The numbers $k_1$ and $k_2$ can be found as follows. Let $u_1=a_{11}x+a_{12}y$, $u_2=a_{21}x+a_{22}y$, and $A=\|a_{ij}\|$, $1\le i,j\le 2$. Since $\{u_1,u_2\}$ is a basis of the group $G$, there exist integers $b_{ij}$, $1\le i,j\le 2$ such that $x=b_{11}u_1+b_{12}u_2$ and $y=b_{21}u_1+b_{22}u_2$. It follows $BA=I$, where $B=\|b_{ij}\|$, $1\le i,j\le 2$, the matrix $A$ is invertible.
Cauchy-Binet formula implies that if $M$ is an integer $n\times n$ matrix and $A$ and is an invertible $n\times n$ integer matrix then matrices $M$ and $MA$ have the same divisors $d_1,\dots, d_k$, where $d_i$ is the greatest common divisors of minors of $i$-th order of the matrix.
Since $$\begin{pmatrix}k_1 & 0\\ 0 & k_2\end{pmatrix} A=\begin{pmatrix}2 & 1\\ 2 & -3\end{pmatrix},$$ $k_1=\gcd (2,1,2,-2)=1$ and $k_1k_2=\gcd\det \begin{pmatrix}2 & 1\\ 2 & -3\end{pmatrix}=8$.
References
[Kur] A. G. Kurosh, Group theory, 3nd ed., Nauka, Moskow, 1967. (in Russian)