Maschke's theorem
Maschke's theorem is the structural cornerstone that makes finite group representation theory so well-behaved: over the right field, you can never get stuck with a representation that has an invariant subspace but refuses to split off. Everything decomposes. The proof idea is a charming averaging trick: take any complement, then average it over the group to smear it into an invariant one.
The precise statement: let G be a finite group and k a field whose characteristic does not divide |G| (for instance any field of characteristic 0, or a finite field of suitable characteristic). Then every finite-dimensional representation of G over k is completely reducible — equivalently, every G-invariant subspace has a G-invariant complement, and equivalently the group algebra k[G] is a semisimple ring.
The hypothesis on the characteristic is essential, not cosmetic. If char(k) = p divides |G|, the averaging operator (1/|G|) sum rho(g) cannot be formed (you would divide by zero), and indeed semisimplicity fails: this is the gateway to modular representation theory, where representations have genuine non-split extensions and the clean direct-sum picture collapses.
To split off an invariant W from V, pick any linear projection p onto W and form the averaged projection p_0 = (1/|G|) sum over g of rho(g) p rho(g)^{-1}. Then p_0 is G-equivariant and projects onto W, so ker(p_0) is the invariant complement.
The averaging trick: this division by |G| is exactly what the characteristic hypothesis protects.