Let $G$ be a group of permutations of a finite set $X$. By the augmentation submodule of $\mathbb CX$, I mean the set of vectors whose coefficients sum to $0$. It is easy to show via character theory that the augmentation submodule is irreducible iff $G$ is doubly transitive. Does anybody know a proof using only the definitions of doubly transitive and irreducible representation and avoiding character theory and the orthogonality relations? It is known (I learned this from Peter Cameron, but don't know a good reference) that replacing $\mathbb C$ by $\mathbb R$ characterizes double homogeneity. So the field being algebraically closed is somehow important.

The motivation for this question comes from trying to understand the relationship between double transitivity for transformation monoids and irreducibility of the augmentation submodule. Character theory for monoids is harder to apply and so an answer to my question may provide some insight.