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Feb
22 |
comment |
Is every finite group a group of “symmetries”?
Thanks! It seems like it would be difficult, in the equivariant embedding, to guarantee that the image has no accidental symmetries (i.e. its isometry group is no larger than G). If one knew this, it would give a sort of perverse answer to the original question: find a hyperbolic surface X (with relatively very large genus) whose isometry group is G, and embed X into some R^n such that the isometries of its image are exactly G. |
Feb
22 |
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Is every finite group a group of “symmetries”?
This construction shows that a finite group G of order n can be realized as exactly the isometries of a convex polyhedron in R^n. Now, suppose m is the least positive integer such that G has a faithful representation in GL(m,R). Typically m is much smaller than n. I am curious whether it is possible to use your strategy to produce a polyhedron in R^m, whose symmetries are exactly the elements of G, considered as elements in GL(m,R). If that could be done, it would be a solution to the original problem with minimal dimension. |
Feb
22 |
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Is every finite group a group of “symmetries”?
This is a really nice argument. I like that it is effective: given an explicit description of a group, one could easily find the vertices of the polyhedron P. It's even almost visualizable. |
Feb
22 |
accepted | Is every finite group a group of “symmetries”? |
Feb
19 |
awarded | Commentator |
Feb
19 |
comment |
Is every finite group a group of “symmetries”?
Hee hee, I'm glad it troubled your sleep! I have an intuitive feeling that the inner product has too few degrees of freedom to kill symmetries as flexibly as needed. But I can't back up that feeling with an actual proof. It would be lovely if one could find a clean, representation theoretical answer to this question! |
Feb
19 |
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Is every finite group a group of “symmetries”?
Neat! Do you know if there are analogous results in higher dimensions? A refinement of the original question, (assuming that previous commenters and I are all correct, and the answer is yes), would be to find lower and upper bounds for the integer n, as a function of the order of the group (or some other data from the group). This would be sort of analogous to embedding theorems for manifolds. If the permutohedron argument suggested by Mariano S-A works, we would have the order of the group as an upper bound. |
Feb
16 |
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Is every finite group a group of “symmetries”?
@RB: Do you have a reference for every finite group being the automorphism group of a Riemann surface? I seem to recall hearing this as folklore, but it occurs to me that I have no idea how it is proved. That suggests another question: is there an "equivariant" embedding theorem for manifolds with symmetries? I.e. if a hyperbolic surface X has nontrivial automorphisms, is there an embedding of X in R^n for some n, such that every automorphism of X is induced by an isometry of R^n preserving the image of X? |
Feb
16 |
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Is every finite group a group of “symmetries”?
@RB and KO'B: I was aware of the variants you mention, and there are certainly more one can cook up. What I wanted to say to the non-mathematician is the most literal generalization of symmetry in the colloquial sense: that every finite group is the isometry group of some polytope in R^n. |
Feb
16 |
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Is every finite group a group of “symmetries”?
@kakaz: the conditions 1 and 2 you say should be added to the original question are, if I understand them correctly, contained in the word "isomorphic". Reid B and Anton G's answers above show that the answer is yes, with 1 and 2 as requirements. |
Feb
16 |
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Is every finite group a group of “symmetries”?
@Mariano S-A: Thanks, I didn't know about the permutohedron! I'll have to check details, but I think your suggestion answers the second part of my original question, giving a simple construction for the required convex set. Just give the group G as a subset of S_n, and then truncate the permutohedron appropriately to restrict its symmetries to the subgroup. |
Feb
15 |
awarded | Critic |
Feb
15 |
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Is every finite group a group of “symmetries”?
You did not read the whole question. As you mention, any finite group is a subgroup of a symmetric group; answering the question for the symmetric group would answer it for all finite groups. However, I was not looking to visualize the group as symmetries of "some kind of graph" (which is straightforward), but as isometries of a convex set in R^n. Note that the previous commenters have given good answers. |
Jan
14 |
awarded | Nice Question |