From: edcjones@access2.digex.net (Edward C. Jones) Newsgroups: sci.math Subject: Finite Groups of Rotations Date: 11 Oct 1995 20:07:57 GMT What are the finite subgroups of the group of rotations in R^n? For which finite sets of points on the n-sphere do there exist a finite group of rotations, G, such that: - Each of the rotations maps the set of points onto itself. - If a and b are two of the points, there is an R in G so that R(a) = b. - The points do not lie in a plane of dimension n-1 or less. Thanks, Ed ============================================================================== From: hawthorn@waikato.ac.nz Newsgroups: sci.math Subject: Re: Finite Groups of Rotations Date: 13 Oct 95 11:08:15 +1300 In article <45h86t\$bfr@news4.digex.net>, edcjones@access2.digex.net (Edward C. Jones) writes: > What are the finite subgroups of the group of rotations in R^n? As far as I know, this problem has not been solved in general. Complete solutions are known for n=1,2,3,4, and a few other small values. (my reference was published in 1985, so I might not be up with the current state of play on this question) The problem however has been completely and elegantly solved for subgroups of the orthogonal group which are generated by reflections. Such groups are called Coxeter groups, and they have been completely classified. Indeed this classification is very important, and is the basis for other famous classification theorems in other branches of mathematics. If you want to read about this, look up the book Finite Reflection Groups, by Grove and Benson Springer 1985 which I recommend as being excellently written and quite readable, and which lays out the complete classification in a rather nice way. > For which finite sets of points on the n-sphere do there exist a > finite group of rotations, G, such that: > - Each of the rotations maps the set of points onto itself. > - If a and b are two of the points, there is an R in G so that > R(a) = b. > - The points do not lie in a plane of dimension n-1 or less. This problem is equivalent to the first - such sets are just orbits of finite rotation groups. > Thanks, > Ed ============================================================================== Newsgroups: sci.math From: dik@cwi.nl (Dik T. Winter) Subject: Re: Finite Groups of Rotations Date: Fri, 13 Oct 1995 01:02:59 GMT In article <1995Oct13.110815.41129@waikato.ac.nz> hawthorn@waikato.ac.nz writes: > If you want to read about this, look up the book > > Finite Reflection Groups, by Grove and Benson > Springer 1985 > Or Coxeter's "Regular Polytopes"; reprinted by Dover. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924098 home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl ============================================================================== From: chenrich@monmouth.com (Christopher J. Henrich) Newsgroups: sci.math Subject: Re: Finite Groups of Rotations Date: Thu, 12 Oct 1995 21:50:32 -0400 In article <45h86t\$bfr@news4.digex.net>, edcjones@access2.digex.net (Edward C. Jones) wrote: > What are the finite subgroups of the group of rotations in R^n? > > For which finite sets of points on the n-sphere do there exist a > finite group of rotations, G, such that: > - Each of the rotations maps the set of points onto itself. > - If a and b are two of the points, there is an R in G so that > R(a) = b. > - The points do not lie in a plane of dimension n-1 or less. > > Thanks, > Ed I believe the classic reference for this question is _Regular Polytopes_ by H. S. M. Coxeter. It is in print, in a sturdy trade paperback edition, for a very reasonable price, from Dover Press. I am not connected with Dover except as a lifelong fan. Regards, Chris Henrich ============================================================================== From: rusin@washington.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Finite Groups of Rotations Date: 13 Oct 1995 06:33:38 GMT In article <45h86t\$bfr@news4.digex.net>, Edward C. Jones wrote: >What are the finite subgroups of the group of rotations in R^n? Depends how you ask the question. If n is allowed to vary, the answer is "all finite groups". If n is assumed to be large but fixed, you can still assume the answer is going to be very messy, since the union of the answers for the various n will include all groups. Actually it's worse than that: it is true that you will sometimes have two subgroups of O(n,R) which are isomorphic as abstract groups but not conjugate (so that as groups of symmetries they are essentially distinct). To a group theorist, this is just the question, what groups have a faithful real representation of degree n? If (as in your other question) you want to assume the group is "essentially" of degree n (not hitch-hiking in from a lower dimension) then you are looking for the group to have an irreducible such representation. It is true that for any n there exists a constant f(n) such that all the finite subgroups of O(n, R) have an abelian subgroup of index at most f(n). The same sentence is true if we insert the word "normal" before abelian. Moreover, the abelian subgroups of O(n,R) obviously have a bounded rank. So for any n an answer may be given which describes all the finite subgroups of O(n, R) up to the resolution of a family of extension problems. I'm not sure for what values of n this has been carried out (probably more than merits doing...) >For which finite sets of points on the n-sphere do there exist a >finite group of rotations, G, such that: > - Each of the rotations maps the set of points onto itself. > - If a and b are two of the points, there is an R in G so that > R(a) = b. > - The points do not lie in a plane of dimension n-1 or less. So you want the points to be an orbit under the action of a G. Your last condition is just the condition that (if the representation is reducible) that the points not lie in some invariant subspace. Seems to me that up to rotations of the sphere the only way to get distinct orbits is to have distinct stabilizers. So the way to get the sets of points you describe is: (1) list all the finite subgroups G of O(n,R); (2) find all the subgroups H of G which are stabilizers of a point (this is usually done by examining the fundamental domain of G's action); (3) display an orbit of G/H. Other posters are mentioning the reflection groups. These provide _some_ of the finite subgroups of O(n,R), but there are others too, so I'm not sure what the point is. dave