A Crystallographic Group is a discrete group of isometries acting on the n-dimensional euclidean space $\mathbb{R} ^n $ with compact fundamental domain.
A lattice is a crystallographic group which consists only of translations.
We have a theorem that says that every crystallographic group contains n linearly independent translations.
Now, we take $L$ to be the subgroup of an n-dimensional crystallographic group $G$ consisting of all pure translations in G.
Does the first theorem I stated implies $L$ must be a lattice?
Thanks a lot !