# Dieudonné and generators of the orthogonal group

Let $k>0$ be a positive integer and $n=4k.$ A special case of a result of Dieudonné is that every element $g$ of the orthogonal group in $n$ variables over the rational numbers $$G=O(n,\mathbb{Q})$$ is a product of at most $n$ elements of order $2$ of $G$.

We want to add some extra condition on $g$ (e.g., the following one)

Question: Can the result be extended to the subgroup of doubly stochastic elements $g$ of $G$.

More precisely:
Let $g \in G$ be such that $g$ is doubly stochastic, that is, the sum of the entries in any fixed row (or column) of $g$ always equal $1$. Assume that $g$ has order $>2$ (e.g., $g$ has infinite order) in $G.$ Note also that the identity has order $1$ by hypothesis.

There exists a decomposition

$$g =g_1 g_2 \cdots g_n$$ with $g_i \in G$ such that

(a) either $g_i$ is equal to the identity or $g_i$ has order $2.$

and

(b) at least one of the $g_i,$ that is not equal to the identity, is also doubly stochastic.

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Given that the identity is doubly stochastic, your conditions can be trivially satisfied (given the theorem you quoted). Could you change them to be what you want? Perhaps you don't want the condition in (a) that the $g_i$ may be the identity? – Noah Stein Jan 11 '11 at 17:37
@Noah: You are right. Edited. – Luis H Gallardo Jan 11 '11 at 18:57
What about to forget the positivity in the doubly stochastic saga ?. What I want is just to understand what are really the conditions implied on the $g_i$'s by the theorem. – Luis H Gallardo Jan 21 '11 at 14:18