 Solution

For finite nonempty $\Omega$, let $e_{\omega }$ be the basis vectors of the permutation representation. PutThe line $\mathbb{C}u$ is fixed by $G$, and $V$ is the augmentation subrepresentation of a permutation representation. Since $|\Omega |\ne 0$ in $\mathbb{C}$, they have zero intersection and their dimensions add to $|\Omega |$. Hence $\boxed{\mathbb{C}\Omega \cong {\mathbb{C}}_G\oplus V}$.

The printed formulation does not explicitly require $\Omega$ to be finite. For an infinite nonempty $G$-set, interpret $\mathbb{C}\Omega$ as finite-support vectors. Choose one finite orbit $O$ (all orbits are finite because $G$ is finite), replace $u$ by ${\sum }_{\omega \in O}{e}_{\omega }$, and use the same augmentation kernel. This again splits off a trivial summand. The nonempty-set assumption is needed; the empty permutation module contains no trivial summand.