This is a timeline page for major breakthroughs in the classification of finite simple groups.
- 1955: Brauer and Fowler published On a group of even order which was a first step towards the odd-order theorem.
- 1960: Suzuki came up with a sporadic simple group, now called the Suzuki group. This was the first sporadic simple group discovered since 1873, and shattered Burnside's conjecture that the Mathieu groups (1860-73) were the only sporadic simple groups.
- 1961: Feit and Thompson proved the odd-order theorem, stating that every finite group of odd order is solvable. Equivalently, there is no simple non-Abelian group of odd order. This paper was a landmark not only for the value of the result but also for the complexity of the proof: it was 255 pages long.
- 1973: The last of the sporadic simple groups, viz the Monster, was predicted by Fischer and Greiss.
- 1980: The Monster was constructed by Greiss as the automorphism group of a certain algebra. This completed the proof of the Classification (it had already been shown that there are no more sporadic simple groups).
- 2004: Aschbacher fixed a hole in the earlier proof, which involved some work with the quasithin case. The new proof obtained was a second-generation classification proof, which was somewhat simplified because of hindsight/foresight.