Classification of Finite Simple Groups - Statement of The Classification Theorem

Statement of The Classification Theorem

Theorem. Every finite simple group is isomorphic to one of the following groups:

• A cyclic group with prime order;
• An alternating group of degree at least 5;
• A simple group of Lie type, including both
  • the classical Lie groups, namely the simple groups related to the projective special linear, unitary, symplectic, or orthogonal transformations over a finite field;
  • the exceptional and twisted groups of Lie type (including the Tits group).
• The 26 sporadic simple groups.

The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups (and their action on other mathematical objects) can sometimes be reduced to questions about finite simple groups. Thanks to the classification theorem, such questions can sometimes be answered by checking each family of simple groups and each sporadic group.

Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups. The completed proof of the classification was announced by Aschbacher (2004) after Aschbacher and Smith published a 1221 page proof for the missing quasithin case.