 Higman criterion for finite representation type of a group algebra

For a finite group $G$ and a field $k$ of characteristic $p$, the group algebra $kG$ has finite representation type exactly when a Sylow p-subgroup of $G$ is cyclic. Restriction and induction reduce the assertion to a Sylow p-subgroup, using projectivity detected on a subgroup of invertible index. A cyclic p-group has only the finitely many indecomposable modules for a cyclic p-group.^[ref]

A noncyclic finite p-group has a quotient group $C_p\times C_p$. Here is an infinite collection of indecomposable modules that works even over a finite field. On $U\oplus W$ with $U=W=k^n$, let $x=g-1$ send $U$ identically onto $W$ and kill $W$, and let $y=h-1$ send $U$ onto $W$ by one Nilpotent Jordan block $J_n$ and kill $W$. All products of $x,y$ vanish, so the defining group algebra relations hold. A commuting endomorphism has block form $\left(\begin{array}{cc}A& 0\\ C& A\end{array}\right)$ with $A{J}_n=J_{n}A$. The quotient by the square-zero ideal of blocks $C$ is $k[J_n]\cong k[t]/(t^n)$, a local ring. The entire endomorphism ring is therefore local, so the module is indecomposable. Their unbounded dimensions $2n$ give infinitely many isomorphism classes over every $k$, rather than relying on an infinite scalar parameter set.