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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2$\begingroup$ What does asymmetry mean? For my understanding "symmetry" is the same as "group action on a set"... $\endgroup$– Abel StolzCommented Mar 30, 2011 at 9:39
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1$\begingroup$ A decent working definition of 'asymmetric' is that an object has trivial automorphism group. For instance almost all finite graphs are asymmetric in this sense. $\endgroup$– Colin ReidCommented Mar 30, 2011 at 9:40
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$\begingroup$ This question is far below the research-level expected on MO. Its most serious flaw is its lack of any sort of definition of the term 'asymmetry'. I've voting to close. Please read the FAQ for a description of the sort of questions that are suitable. $\endgroup$– HJRWCommented Mar 30, 2011 at 11:47
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2$\begingroup$ The people giving answers are trying to guess what the question means, but shouldn't have to - it's the questioners job to make clear what the question means. Voting to close. $\endgroup$– Gerry MyersonCommented Mar 30, 2011 at 12:19
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$\begingroup$ Maybe define it with respect to a specific action of a group as in lack of reflection symmetry for certain points in a function, i.e. absence of a certain symmetry. $\endgroup$– bonifCommented Aug 28 at 17:56
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2 Answers
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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.
answered Mar 30, 2011 at 9:43
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"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.
answered Mar 30, 2011 at 9:58