From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: subgroups of SO(n) acting on S^{n-1}
Date: 29 Jan 2000 07:39:30 GMT
Newsgroups: sci.math
Summary: [missing]

In article <86bemu$8i9$1@nnrp1.deja.com>,
David Bernier  <bernier@my-deja.com> wrote:
>I thought of a few solids and their symmetry groups:
>
>(a) Regular pyramid with base being a regular n-gon for n=>4.
>(b) Two identical n-gon base pyramids put base to base,
>(c) The symmetry groups for the 5 Platonic solids (up to five
>    groups).
>(d) We can add all subgroups of cases (a), (b) and (c).
>
>Are there any other finite subgroups of SO(3)? [up to
>equivalence modulo elements of SO(3)]  Or,
>how can one classify all finite subgroups of SO(3)?

Keep in mind that there are several questions here. For example, how many
groups of order 2 are there in  O(3)? There are clearly infinitely many,
since we may rotate by  pi  around any axis. Perhaps you mean to consider
some groups "equivalent"; fine, but which? Are these rotations equivalent to
a reflection? -- they are isomorphic groups (all groups of order 2 are 
isomorphic). The usual answer is, "classify the finite subgroups of  O(n)
and  SO(n)  up to conjugacy".

The finite subgroups of  SO(3)  up to conjugacy are
(a) The cyclic groups of order n (any positive integer  n)
(b) The dihedral groups of order 2n (when n=1 this is cyclic, else not)
(c) The alternating group  A_4  of order 12 (rotations of the tetrahedron)
(d) The symmetric group S_4 of order 24 (rotations of cube and octahedron)
(e) The alternating group A_5 of order 60 (rotations of icos. and dodec.)
Observe that this collection of groups is already closed under the taking
of subgroups.

This ought to be in Coxeter's Geometry book(s). There's a little more
variety if you ask for finite subgroups of  O(3). One may in fact
classify the finite subgroups of  SO(n)  for any  n  given
sufficient patience. As in the case n=3 there may be infinitely many
such groups but there will be an abelian normal subgroup of bounded index
and bounded rank.

dave
Group theory: index/20-XX.html