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

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I've finally found my copy of Conway & Smith (which I highly recommend, although you could spend a lifetime trying to absorb all the goodies therein). At last I know what Fred is talking about. The proof is flawless, if laconic. ----------------------------------------------------------------------------------------------------- Lemma 3 (page 6, "On Quaternions and Octonions", John Conway and Derek Smith, A.K. Peters, 2003): Every element g of O(n) that fixes a k-dimensional subspace can be written as a product of at most n-k reflections. ----------------------------------------------------------------------------------------------------- [Note: O(n) is the group of all isometries of R^n that fix the origin. This book calls it GO_n, but the rest of the world calls it O(n) or O_n.] ----------------------------------------------------------------------------------------------------- Proof as given in the book: Take a vector v that is *not* fixed by g, say v -> w. Then reflection in v-w restores w to v while fixing any vector u fixed by g. ------------------------------------------------------------------------------------------------------ Explication du texte: We're doing induction on the codimension n-k. First assume n-k = 0. Then g fixes all of R^n and we're done, having used 0 reflections. Now assume Lemma proved for all codimensions < n-k. Let r denote the reflection in v-w. Now compose r with f to get rf, satisfying rf(v) = r(w) = v. It's easy to check that rf fixes the same k-dimensional subspace fixed by f. But rf also fixes the subspace generated by v, so the dimension of rf's fixed subspace greater than k and its codimension is less than n-k. Hence by the induction hypothesis, rf is the product of at most n-k-1 reflections: rf = r_1 r_2 . . . r_q for q <= n-k-1. Applying r on the left of each side of this equation, we get (since r^2 = identity) that f is the prouct of at most n-k reflections. qed ------------------------------------------------------------------------------------------------------- --Dan