Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 1 / b / Solution

For a finite-dimensional module over $A$, the radical of a module satisfies $J(M)=J(A)M$. Henceand induction gives $J^r(U\oplus V)=J^r(U)\oplus J^r(V)$ for every $r$. A simple submodule of a direct sum projects into semisimple submodules of each summand, and equivalentlyThus the same argument, or induction through the defining quotients, givesThis is radical and socle series of a direct sum.

Now let $G$ be a finite $p$-group and let $k$ have characteristic $p$. The group algebra of a p-group in characteristic p is local, with unique simple module $k$. The socle of its regular module iswhich is one-dimensional. If $kG=U\oplus V$ with both summands nonzero, finite length gives nonzero socles for $U$ and $V$, and the direct-sum identity would make $\mathrm{Soc}(kG)$ at least two-dimensional. Therefore