---
tag: 06J6P03MG7
kind: statement
role: theorem
name: Classification of finite simple groups
aliases: [enormous-theorem]
msc: [20D05]
uses: [0C093D1T4M, 0T13HM377F, 0B85Q41RP6]
sources:
  - url: https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
    locator: Statement of the classification theorem
same_as:
  wikidata: Q1340623
  wikipedia: Classification of finite simple groups
---

## Statement

Each finite [[simple group|0YMYMNQ0QZ]] is isomorphic to one of the following groups:

1. a [[cyclic group|0KVGH0J5B3]] $\mathbb{Z}/p\mathbb{Z}$ of prime order $p$;
2. an [[alternating group|0Z5905Q73D]] $A_n$ with $n \geq 5$;
3. a group of Lie type;
4. one of $26$ sporadic groups.

## Remarks

The groups of Lie type form infinite families, for example the groups $\mathrm{PSL}_n(\mathbb{F}_q)$, with a few exceptions for small $n$ and $q$.
Some authors count the Tits group as a $27$th sporadic group, because it is not strictly of Lie type.
The largest sporadic group is the Monster group.
The proof consists of tens of thousands of pages in several hundred journal articles.

## History

Daniel Gorenstein announced in 1983 that the classification was complete, but this was premature.
The gap was the case of the quasithin groups.
Michael Aschbacher announced the completed proof in 2004, after Aschbacher and Stephen Smith published a proof of $1221$ pages for this case.

## Explanations

### L1 human

The theorem lists all the finite simple groups.
Most of them come in a few infinite families, and only $26$ are special cases.
Since each finite group is built from simple groups, this list is a map of all the building blocks.