  Given an odd prime number $p$ and a Coxeter group $W$ such that the order of
the product $st$ is prime to $p$ for every Coxeter generators $s,t$ of $W$, we
prove that the $p$-local homology groups $H_k(W,\mathbb{Z}_{(p)})$ vanish for
$1\leq k\leq 2(p-2)$. This generalize a known vanishing result for symmetric
groups due to Minoru Nakaoka.

Algebraic Topology;Group Theory