Theorem 1

(W, S) is a Coxeter system if and only if it satisfies the exchange condition (E) of no. 5.

Lemma 3 of no. 4 shows that any Coxeter system satisfies (E).

Conversely, suppose that (E) is satisfied. Let G be a group and f a map from S to G such that $(f(s)f(s'))^m = 1$ whenever s and $s'$ belong to S and $ss'$ is of finite order m. By Prop. 5, there exists a map g from W to G such that

whenever $w = s_1 \ldots s_q$ is of length q. To prove that (W, S) is a Coxeter system, it suffices to prove that g is a homomorphism, which is a consequence of the formula since S generates W. By Prop. 4 of no. 5, only two cases are possible:

a) $l(sw) = l(w) + 1$: if $(s_1, \ldots, s_q)$ is a reduced decomposition of $w$, then $(s, s_1, \ldots, s_q)$ is a reduced decomposition of $sw$, hence (21).

b) $l(sw) = l(w) - 1$: put $w' = sw$; then $w = sw'$ and $l(sw') = l(w') + 1$. By a), $g(w) = f(s)g(sw)$ and hence $f(s)g(w) = g(sw)$ since $(f(s))^2 = 1$.