Regular representation

The left regular representation acts on the basis $(e_h{)}_{h\in G}$ by $\rho (g)e_h=e_{gh}$. It is faithful.

If ${\rho }_1,\dots ,{\rho }_k$ are the complex irreducible representations of a finite group $G$ and $n_j=\dim {\rho }_j$, decomposition of the regular representation gives

By Maschke's theorem, the complex regular representation of a finite group is a direct sum of irreducibles. Its kernel is the intersection of their kernels. If the group is simple, each kernel is either trivial or the whole group; faithfulness of the regular representation therefore forces at least one irreducible constituent to be faithful.