The algebraic definition of a crystallographic group goes as follows:
If a group $\Gamma$ fits into a short exact sequence $$0 \to \mathbb{Z}^n \overset{i}{\to} \Gamma \overset{p}{\to} G \to 1$$ such that $i \left(\mathbb{Z}^n \right)$ is maximal abelian in $\Gamma$ and $G$ is finite, then $\Gamma$ is a crystallographic group. In this case, $n$ is called the dimension of $\Gamma$ and $G$ is called the holonomy group of $\Gamma$.
Why is the dimension of a crystallographic group unique? I guess it has something to do with $i \left(\mathbb{Z}^n \right)$ being maximal abelian, but I can't figure it out.
Hint. Your question is equivalent to proving that $\mathbb{Z}^n$ cannot contain $\mathbb{Z}^m$ with finite index, unless $m=n$.