Two-sided regular representation decomposition
= Two-sided regular representation decomposition
Let $G\times G$ act on $\mathbb C G$ by $(g,h)x=gxh^{-1}$. Its character is zero unless $g$ and $h$ are conjugate, in which case it equals $|C_G(g)|$. Character column orthogonality gives the same character for
$$
\bigoplus_i\operatorname{Hom}_{\mathbb C}(V_i,V_i),
$$
where the $V_i$ run through the irreducible complex representations. Maschke's theorem therefore gives an isomorphism of the two representations.