Two responses on the SO(3) question. Also, many about Simpson,
Slack, Steinbrink, Smith, and Hanke. That leaves the following:
> Jacob Davis jacob.d@mail.utexas.edu
> Jim Krevitt krevitt@attbi.c
> Pelle Salomonsson salomonsson@mathematik.su.se
> Vladimir Jojic vjojic@santafe.edu
> Marc Guastavino marc@texnet.qc.ca
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Subject: Re: question abt SO(3)
Date: Wed, 24 Jul 2002 09:54:30 -0400 (EDT)
From: "Douglas C. Ravenel"
> Date: Tue, 23 Jul 2002 13:54:23 -0400 (EDT)
> From: Glen Takahara
>
> If U1,...,U5 are 5 distinct elements of SO(3),
> the proper symmetric group, then I am able to
> prove that U1 + ... + U5 = 0 is impossible.
> However, my proof is brute force and I would
> like to know if anyone has some intuition or
> high-level reason why they would expect this
> equality to be impossible.
>
If by SO(3) you mean the 3x3 orthogonal group and by addition you
mean matrix addition, then the result is false. Choose a plane
and consider the 5 matrices corresponding to rotations in that
plane by multiples of 2\pi/5. Their sum is zero by symmetry, or
by explicit calculation in the case where the plane is the
XY-plane.
Douglas C. Ravenel, Chair |918 Hylan Building
Department of Mathematics |drav@math.rochester.edu
University of Rochester |(585) 275-4413
Rochester, New York 14627 |FAX (585) 273-4655
____________________________________________
Subject: Re: question abt SO(3)
Date: Wed, 24 Jul 2002 10:20:50 -0400 (EDT)
From: Walter Neumann
I have never heard the terminology "proper symmetric group" (and neither
mathscinet nor google have either) but assuming SO(3) means the special
orthogonal group (group of orientation preserving symmetries of R^3 fixing
the origin), the statement is surely wrong. It is not even true in the
subgroup SO(2) (the circle group), e.g., Uj= rotation by 2j\pi/5
--walter neumann