TITLE: Quotient of two free abelian groups of the same rank is finite?
QUESTION [10 upvotes]: Let $A,B$ be abelian groups such that $B\subseteq A$ and $A,B$ both are free of rank $n$. I want to show that $|A/B|$ is finite, or equivalently that $[A:B
]$ (the index of $B$ in $A$) is finite.
For example, if $A=\mathbb{Z}^n$ and $B=(2\mathbb{Z})^n$ we have that $[A:B]=2^n$ (correct me if I'm wrong).

REPLY [21 votes]: Let $A\cong B\cong \mathbb{Z}^n$, with $B\subseteq A$.  Then we have the short exact sequence 
$$ 0\rightarrow B\rightarrow A\rightarrow A/B\rightarrow 0.$$
Tensoring with $\mathbb{Q}$ shows $A/B$ has rank $0$, so is torsion.