In an informal sense, groups are related to symmetry. I was wondering if there are groups that describe some sort of asymmetry. Does anyone know of such groups or is asymmetry and groups a contradiction in terms?
Thanks a lot
In an informal sense, groups are related to symmetry. I was wondering if there are groups that describe some sort of asymmetry. Does anyone know of such groups or is asymmetry and groups a contradiction in terms? Thanks a lot 


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Maybe it is really a contradiction: once you have a group acting on some set preserving some structure you have a group homomorphism from your group into the automorphism group of that structure. This is really a very general phenomenon. However, to give you some example which goes perhaps more into your direction: in physical models of solid states you have in crystals a discrete symmetry group of discrete translations by a lattice. Clearly, you would call a crystal "symmetric", historically it is perhaps the origin of all considerations concerning symmetry. But now you perturb the symmetry and spoil your crystal. No lattice acts any more. But if you do that very much you arrive at something similar to a "glas": no obvious symmetry but approximately on a larger scale perhaps, it looks very homogeneous again. In this approximative sense, it might be justifiable to call a glass even translation invariant under all translations. 


If you start with a very symmetric object (for example a sphere), you have a large symmetry group. If you break the symmetry (for example. you color two antipodal points of the above mentioned sphere), the symmetry group becomes smaller (in this case one is left with rotations on the axis through these points and some reflections). The amount that the group is reduced can be understood as some measure of asymmetry (of the sphere with marked points, with regards to the sphere without marked points). Or is this not something you are after? 


"Broken symmetry", rather than complete and utter asymmetry, is an important concept for physicists. See http://en.wikipedia.org/wiki/Symmetry_breaking. I suppose this all goes back to Buridan's ass (http://en.wikipedia.org/wiki/Buridan%27s_ass). But it doesn't fit that well with group theory as mathematicians see it, though obviously descending to a subgroup as symmetry group is a valid conceptual framework. 

