A representation of a finite group \(G\) over a field \(k\) is a finite-dimensional \(k\)-vector space \(V\) together with a homomorphism \(\rho\colon G\to\GL_k(V)\); equivalently, it is a finite-dimensional left module over the group ring \(k[G]\). By Maschke’s theorem, if \(\operatorname{char}k\) does not divide \(\size G\), then every finite-dimensional \(k[G]\)-module is semisimple.
For \(k=\CC\):
Maschke and Schur – semisimplicity, and homomorphisms between irreducible representations.
Characters – orthogonality relations and character tables.