From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Spherical orbits
Date: 9 Aug 2000 00:03:32 GMT
Newsgroups: sci.math
Summary: [missing]

In article <39868E17.70F668D4@club-internet.fr>,
Nicola Sottocornola  <n_sotto@club-internet.fr> wrote:

>let H be a sub-group of O(n). For each P in R^n, his orbit under H is
>the sphere with radius OP.
>Can we say that SO(n) is included in H?

No. For n=4 we may identify  R^n  with the additive group of quaternions.
The group H of unit quaternions acts on  R^n  by left multiplication
and preserves lengths, so that for an individual  P,  the orbit  HP
is contained in the sphere  S  with radius  OP.  On the other hand,
(for  P  not at the origin)  if  v  lies on  S  then  x= v P^{-1} 
carries  P  to  v  and has length  1, i.e. lies in  H; this shows the
orbit  HP  is all of  S.

dim(H)=3  but  dim(SO(4)) = 6.

dave