The notion of the extremal length and the module of families of curves has been
studied extensively and has given rise to a lot of applications to complex
analysis and the potential theory. In particular, the coincidence of the
$p$-module and the $p$-capacity plays an important role. We consider this
problem on the Carnot group. The Carnot group $\mathbb{G}$ is a simply connected
nilpotent Lie group equipped with an appropriate family of dilations. Let
$\Omega$ be a bounded domain on $\mathbb{G}$ and $K_0$, $K_1$ be disjoint
non-empty compact sets in the closure of $\Omega$. We consider two quantities,
associated with this geometrical structure $(K_0,K_1;\Omega)$. Let
$M_p(\Gamma(K_0,K_1;\Omega))$ stand for the $p$-module of a family of curves
which connect $K_0$ and $K_1$ in $\Omega$. Denoting by $\cap_p(K_0,K_1;\Omega)$
the $p$-capacity of $K_0$ and $K_1$ relatively to $\Omega$, we show that
$$M_p(\Gamma(K_0,K_1;\Omega))=\cap_p(K_0,K_1;\Omega)$$.