From a $n$-polygon in $\mathbb{R}^3$ the set of distances between all pairs of vertices are given. (How) is it possible to reconstruct the geometric structure of the polygone? 

Symbolically:
For a set of coordinates $(x_i,y_i,z_i)$ with $i=1,...,n$ we have given 
$r_{k} = (x_i -x_j)^2 + (y_i - y_j)^2 + (z_i - z_j)^2$, for $k = 1,...,n(n-1)/2$
without knowing the map $k\rightarrow (i,j)$ and want to know the set of $(x_i,y_i,z_i)$.

It seems clear, that the distances between the points determine the polygon only up to translations and rotations. So we are seeking to determine only 3$n$-6 degrees of freedom and we have $n(n-1)/2$ equations. As well the coordinates are only determined up to an arbitrary permutation.

I have absolutely now clue as to how address the question, I guess the answer will somehow involve group theory.

It is motivated by the determination of molecular structures using diffraction   
methods. Its a kind of toy-version (all elements the same) of the so called inverse structure problem. I am doing research on this subject, but according to my current knowledge no one ever looked closer at the mathematical basis of the problem. It is usually solved using model structures (You just assume the tape of polygon and fit the structure parameters (coordinates) to the diffraction data.

(I am also clueless about the correct tags for the disciplines that involves, so sorry for wrong guesses in case).