The paper is based on the central idea underlying the so-called direct methods in crystal structure determination by X-ray diffraction. Because of constraints imposed by atomic structure and positive definiteness of the crystal density function, not all triply periodic functions can occur. This implies relations among the absolute values of the Fourier coefficients of the electron density measured in X-ray diffraction experiments. From those relations information about the phases of the Fourier coefficients can be obtained. The aim of the paper is to show that one can also take advantage from the additional restrictions imposed by invariance with respect to the space group G of the crystal (i.e. the group of Euclidean transformations leaving the crystal invariant). Actually, the problem is reduced to that of finite Fourier transforms of a set of G-invariant Fourier coefficients by applying to the crystal periodic boundary conditions, i.e. by taking the finite homomorphic image of the group of lattice translations. Performing the computation by sampling on \(N\times N\times N\) points, for N not too large, one can reduce the computation by a factor that is about the order of the group G and still be able to apply fast Fourier transform algorithms. In the paper the non-realistic case of \(N=5\) is considered as illustration. One of the authors (Mrs. M. Shenefelt) at the Large Scale Computing Center, New York is engaged in the implementation of an algorithm on space group invariant finite Fourier transforms.