In this paper, we study the discreteness criteria for subgroups
of $U(1, n; \mathbf{C})$ in complex hyperbolic space $H^n_{\mathbf{C}}$.
We prove that a nonelementary subgroup $G$ of $U(1, n; \mathbf{C})$
with condition A is discrete if and only if every two generator
subgroup of $G$ is discrete. We also prove that if a nonelementary
subgroup $G$ of $U(1, n; \mathbf{C})$ contains a sequence of
distinct elements $\{g_m\}$ with
$\operatorname{Card}(\operatorname{fix}(g_m) \cap \partial H^n_{\mathbf{C}}) \ne \infty$
and $g_m \rightarrow I$ as $m \rightarrow \infty$,
then $G$ contains a non-discrete, nonelementary two generator subgroup.