Edit (GK): This would also be my first answer, let me add a few details. The Feit-Thompson theorem asserts that every finite group of odd order is solvable. An equivalent formulation is that every simple nonabelian group is of even order. The theorem was proved by Feit and Thompson in 1962,1962. It was conjectured by Burnside by 1911. The theorem plays a crucial role in the classification of finite simple groups. Some parts of the proof were simplified over the years but it remained very hard.
