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An $A$-module is a semisimple module when it is a direct sum of simple modules, equivalently when every submodule has a complementary submodule. The finite-dimensional algebra $A$ is a semisimple algebra when its left regular module ${}_{A}A$ is semisimple.

Let $C_{p^n}=⟨g⟩$, put $N=p^n$, and suppose $\mathrm{char}k=p$. The freshman's dream givesso, with $u=g-1$,The indecomposable modules of a cyclic p-group in characteristic p are preciselyIndeed, a module is a vector space with a nilpotent operator $u$, and its decomposition into Nilpotent Jordan blocks gives these modules. Each $M_r$ is a uniserial module, with unique chainTherefore

Let $U\le {\mathrm{SL}}_2(p)$ be the Sylow p-subgroup of upper unitriangular matrices. It is cyclic of order $p$, generated byRealize $S^r(V_2)$ as the homogeneous polynomials of degree $r$ in $X,Y$, with $u$ acting by $X\mapsto X$ and $Y\mapsto X+Y$. For $0\le r<p$, the only vectors fixed by $u$ are the multiples of $X^r$: successive comparison of the coefficients of $Y^r,Y^{r-1},\dots$ proves this. Thus the nilpotent operator $u-1$ has one-dimensional kernel. Its Jordan normal form therefore has a single block, soThis also follows from the indecomposable modules of a cyclic p-group in characteristic p.

For $r=p-1$, the restriction has dimension $p$ and is the regular $kU$-module, hence is projective. Since $[G:U]$ is prime to $p$, part (b)(ii) makes $S^{p-1}(V_2)$ a simple projective $kG$-module. It is therefore a defect-zero representation and lifts to an ordinary irreducible representation of the same dimension. Consequently

Let $P$ be a Sylow p-subgroup. Since $[G:P]$ is invertible in $k$, every $kG$-module is relatively $P$-projective.

Suppose first that $kP$ has only finitely many indecomposable modules $U_1,\dots ,U_t$. For every indecomposable $kG$-module $M$, decompose ${\mathrm{Res}}_P^{G}M$ into the $U_i$. Relative projectivity makes $M$ a summand of the corresponding finite direct sum of the ${\mathrm{Ind}}_P^G{U}_i$. The Krull–Schmidt theorem leaves only finitely many possible indecomposable summands, so $kG$ has finite representation type.

Conversely, suppose $kG$ has finitely many indecomposables. For an indecomposable $kP$-module $U$, the identity double coset in the Mackey restriction formula shows that $U$ is a direct summand ofDecomposing the induced module into the finitely many $kG$-indecomposables and restricting them shows, again by Krull–Schmidt, that only finitely many $U$ can occur. Thus

If $P$ is cyclic, the indecomposable modules of a cyclic p-group in characteristic p form a finite list. If $P$ is noncyclic, its Frattini quotient has rank at least two and therefore has a quotient $C_p\times C_p$. Inflation preserves indecomposability and nonisomorphism, while $k(C_p\times C_p)$ has infinitely many indecomposable modules. The Higman criterion for finite representation type of a group algebra now gives