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
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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
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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