# Given positions find the symmetry group

Given a finite set of vectors in $$\mathbb{R}^n$$ ($$n=2,3$$), is there any algorithm to find its symmetry group? For example, if the input is {(1,0),(0,1),(-1,0),(0,-1)}, then the output is the dihedral group of eight elements. Any reference could be helpfull for me.

• It's an interesting question. Can I just check if you are wanting your symmetries to be linear? For instance, if the vectors in your example were all translated up $1$, would you want the algorithm to spit out the trivial group or the dihedral group of eight elements? Commented Dec 10, 2019 at 11:16
• Put another way, you are asking for the setwise-stabilizer of a bunch of vectors, but to understand the question we need to know what group you are sitting inside. So, my earlier comment was to check whether you are thinking inside $GL(n,\mathbb{R})$ or $AGL(n,\mathbb{R})$. However, equally, perhaps you are really working inside $O(n)$ (i.e. rotations/ reflections) or $E(n)$ (rigid transformations). Once you know what group you are in, the algorithm you are seeking will consist of examining certain cosets of subgroups and checking intersections. Commented Dec 10, 2019 at 11:28
• A discussion of algorithms which calculate setwise-stabilizers, you could look at this thesis: math.uni-rostock.de/~rehn/docs/diploma-thesis-cs-rehn.pdf Commented Dec 10, 2019 at 11:31
• I presume you are looking for symmetries which preserve the distance between any pair of vectors. Commented Dec 10, 2019 at 11:38
• OK, in that case, the group you are working inside is $O(n)$. Your question is asking for an algorithm to calculate the setwise-stabilizer in $O(n)$ of some bunch of vectors in $\mathbb{R}^n$. The link I gave above describes such algorithms in general. For $n=2$, the group $O(2)$ is particularly simple -- it is a degree 2 extension of the circle group, so all subgroups are cyclic or dihedral, and so finding a setwise stabilizer is simply a matter of grouping your set of vectors so that they all have the same norm, and then checking angles between them. Commented Dec 11, 2019 at 10:20