nLab classification of finite simple groups

There are 18 countably infinite families and 26 sporadic finite simple groups. In slightly more detail, a finite simple group is one of the following

1. A group of prime order
2. An alternating group $A_n$ for $n\geq 5$
3. A group of Lie type over a finite field
4. One of the 26 sporadic finite simple groups.

The original ‘proof’ fills 500 journal articles. An updated, self-contained proof is in the process of being written, and it is estimated that it will be 5000 pages long. As of 2018 seven volumes had been published, out of an expected 11.

For now see the Wikipedia page.

An original conceptual insight into the classification of finite groups from the point of algebraic geometry (involving embeddings into algebraic groups) has been recently achieved in an award-winning article

• Michael J. Larsen, Richard Pink, Finite subgroups of algebraic groups, J. Amer. Math. Soc. 24 (2011), 1105-1158 doi

category: algebra