  Let $S_{1}=\left\{F_t\right\}_{t\geq 0}$ and $S_{2}=\left\{G_t\right\}_{t\geq
0}$ be two continuous semigroups of holomorphic self-mappings of the unit disk
$\Delta=\{z:|z|<1\}$ generated by $f$ and $g$, respectively. We present
conditions on the behavior of $f$ (or $g$) in a neighborhood of a fixed point
of $S_{1}$ (or $S_{2}$), under which the commutativity of two elements, say,
$F_1$ and $G_1$ of the semigroups implies that the semigroups commute, i.e.,
$F_{t}\circ G_{s}=G_{s}\circ F_{t}$ for all $s,t\geq 0$. As an auxiliary
result, we show that the existence of the (angular or unrestricted) $n$-th
derivative of the generator $f$ of a semigroup $\left\{F_t\right\}_{t\geq 0}$
at a boundary null point of $f$ implies that the corresponding derivatives of
$F_{t}$, $t\geq 0$, also exist, and we obtain formulae connecting them for
$n=2,3$.
