Let $G,H$ be free abelian groups such that $H\leq G$ and $rank(G)=rank(H)=n$.
Let $\alpha,\beta$ be bases for $G,H$ respectively and $M\in Mat_{n\times n}(\mathbb{Z})$ be the base changing matrix from $\beta$ to $\alpha$. (i.e. $M=[id]_\beta^{\alpha}$)
How do I prove that $|G:H|$ is finite and $|G:H|=|det(M)|$?