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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 !

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What is a "pure translation"? –  rschwieb May 21 '12 at 16:41
    
Is a pure translation is a translation, that looks like a tautology to me. Except that you must show that $n$ independent translations in dimension $n$ imply compact fundamental domain, but that is true as well. –  Marc van Leeuwen May 21 '12 at 16:51

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