Classification of finite simple groups
ID: classification-of-finite-simple-groups
Classification of finite simple groups by 
Ciro Santilli 37 Updated 2025-06-17 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-06-17 +Created 1970-01-01Ciro Santilli is very fond of this result: the beauty of mathematics.
How can so much complexity come out from so few rules?
How can the proof be so long (thousands of papers)?? Surprise!!
And to top if all off, the awesomely named monster group could have a relationship with string theory via the monstrous moonshine?
The classification contains:
- cyclic groups: infinitely many, one for each prime order. Non-prime orders are not simple. These are the only Abelian ones.
- alternating groups of order 4 or greater: infinitely many
- groups of Lie type: a contains several infinite families
- sporadic groups: 26 or 27 of them depending on definitions
Simple Groups - Abstract Algebra by Socratica (2018)
Source. Good quick overview.
The Classification of Finite Simple Groups is a monumental result in the field of group theory, specifically in the area of finite groups. It establishes a comprehensive framework for understanding the structure of finite simple groups, which are the building blocks of all finite groups in a manner akin to how prime numbers function in number theory.
New to topics? Read the docs here!