Er, guys, would one of you be so kind as to state what theorem is being discussed here? Thanks, Dan ------------------------------------------------------- Gareth wrote: << On Sunday 08 January 2006 14:41, Fred Lunnon wrote:
Which reminds me that I recently acquired a copy of the august tome J.H.Conway, D.A Smith "On Quaternions and Octonions" and promptly became mired at Lemma 3 on page 6. I can prove this result easily, but I'm hanged if I can follow their (3 line) proof --- can another reader clarify? Fred Lunnon
I'll have a go. Let a be any element of GO(n) fixing a k-dimensional subspace. If it's the identity, we're done. Otherwise, pick something it doesn't fix; say it takes v to w. Let b be reflection in v-w. Then ba is still in GO(n); it fixes all the same things a did; and it also fixes v. So it fixes (at least) a (k+1)-dimensional subspace, so by induction on n-k we're done. How do we know that ba fixes all the same things a did? Because anything a fixes is orthogonal to v-w (Why? Because an isometry preserves inner products; so if au=u and av=w then (u,v) = (au,av) = (u,w), so (u,v-w)=0.) and is therefore fixed by b as well as by a. Is that any better?