Indecomposable modules of a cyclic p-group in characteristic p
= Indecomposable modules of a cyclic p-group in characteristic p
{title2=$kC_{p^n}$}
Let $C_{p^n}=\langle g\rangle$ and let $k$ have characteristic $p$. With $u=g-1$,
$$
kC_{p^n}\cong k[u]/(u^{p^n}).
$$
Its finite-dimensional indecomposable modules are
$$
M_r=k[u]/(u^r),
\qquad 1\leq r\leq p^n.
$$
Each is a <uniserial module>, with radical $uM_r$ and one-dimensional socle $u^{r-1}M_r$.