Theorem 1

Suppose that the group G is finite. Let M be a K[G]-module, and let N be a K[G]-submodule of M. Suppose that N is a direct factor of the K-module M and that $|G|$ is invertible in K. Then N is a direct factor of the K[G]-module M.

Let $p$ be a K-linear projector in M with image N. We define an endomorphism $q$ of the K-module M by setting

for every $x\in M$. Since N is stable under the action of G and $p$ induces the identity on N, we see that $q$ sends M into N and induces the identity on N.

The K-module M is therefore the direct sum of the image N of $q$ and the kernel of $q$. We have $g q(x) =q(gx)$ for every $x\in M$ and $g\in G$, so that the kernel of $q$ is a K[G]-submodule of M. This proves that N is a direct factor of the K[G]-module M.