# nLab classification of finite simple groups

group theory

### Cohomology and Extensions

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.