Finite representation type of a group algebra

A group algebra $kG$ has finite representation type when it has only finitely many isomorphism classes of finite-dimensional indecomposable modules.

If $k$ has characteristic $p$, then $kG$ has finite representation type exactly when a Sylow p-subgroup of $G$ is cyclic. Restriction and induction reduce the property to the Sylow subgroup; a cyclic p-group has the finitely many indecomposables $k[u]/(u^r)$, while a noncyclic p-group has a quotient $C_p\times C_p$ and hence infinitely many indecomposables.