(a) State and prove Schur's lemma over $\mathbb{C}$.

In the remainder of this question we work over $\mathbb{R}$.

(b) Let $G$ be the cyclic group of order 3 .

(i) Write the regular $\mathbb{R} G$-module as a direct sum of irreducible submodules.

(ii) Find all the intertwining homomorphisms between the irreducible $\mathbb{R} G$-modules. Deduce that the conclusion of Schur's lemma is false if we replace $\mathbb{C}$ by $\mathbb{R}$.

(c) Henceforth let $G$ be a cyclic group of order $n$. Show that

(i) if $n$ is even, the regular $\mathbb{R} G$-module is a direct sum of two (non-isomorphic) 1dimensional irreducible submodules and $(n-2) / 2$ (non-isomorphic) 2-dimensional irreducible submodules;

(ii) if $n$ is odd, the regular $\mathbb{R} G$-module is a direct sum of one 1-dimensional irreducible submodule and $(n-1) / 2$ (non-isomorphic) 2-dimensional irreducible submodules.