Let $k$ be an algebraically closed field of characteristic $p>0$. For a loop
$\circlearrowleft$, denote its path coalgebra by $k\circlearrowleft$. In this
paper, all the finite-dimensional commutative Hopf algebras over the sub
coalgebras of $k\circlearrowleft$ are given. As a direct consequence, all the
commutative infinitesimal groups $\mathcal{G}$ with
dim$_{k}$Lie$(\mathcal{G})=1$ are classified.