  The following results are proved: The center of any finite index subgroup of
an irreducible, infinite, non-affine Coxeter group is trivial; Any finite index
subgroup of an irreducible, infinite, non-affine Coxeter group cannot be
expressed as a product of two nontrivial subgroups. These two theorems imply a
unique decomposition theorem for a class of Coxeter groups. We also obtain that
the orbit of each element other than the identity under the conjugation action
in an irreducible, infinite, non-affine Coxeter group is an infinite set. This
implies that an irreducible, infinite Coxeter group is affine if and only if it
contains an abelian subgroup of finite index.
