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.

For $P=C_p\times C_p$, put $x=g-1$, $y=h-1$. On a basis $v,w$, prescribe $xv=w$, $xw=0$, $yv=\lambda w$, $yw=0$. These are valid modules because every product of two radical operators is zero. Each is indecomposable, and an isomorphism preserves the equality $y=\lambda x$, hence preserves $\lambda$. An infinite field therefore gives infinitely many isomorphism classes.

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.