Classification of finite simple groups

The classification theorem states that every finite simple group is cyclic of prime order, an alternating group, a group of Lie type, or one of 26 sporadic groups.

Simple group: A group with no normal subgroups other than the identity subgroup and the group itself. Simplicity is the property that defines the objects in the classification.
Finite group representation theory: The study of group actions by linear transformations on vector spaces. Representations reveal structural constraints used throughout classification proofs.
Camille Jordan: A French mathematician whose work helped establish early foundations of group theory. Jordan’s work on permutation groups helped shape the subject’s early study of simple groups.
Monster group: The largest sporadic simple group, with order approximately 8 × 10^53. The Monster is the most prominent and largest of the 26 sporadic cases.
Finite symmetry group: A finite group describing symmetries that preserve a mathematical or physical structure. The classified groups provide fundamental building blocks for finite symmetry.
Finite group: A group containing finitely many elements. The theorem covers only groups of finite order.
Sylow theorems: Theorems describing subgroups whose orders are the largest prime powers dividing a finite group’s order. Sylow subgroup structure supplies essential local data about finite groups.
William Burnside: A British mathematician who made foundational contributions to finite group theory. Burnside’s work helped develop the framework in which finite simple groups were studied.
Mathieu group: Any of five sporadic simple groups discovered by Émile Mathieu in the nineteenth century. The Mathieu groups were early examples of exceptional simple groups outside the main families.
Jordan–Hölder theorem: A theorem stating that a finite group’s composition factors are unique up to order and isomorphism. The classification identifies the possible simple factors in finite groups.

Linked from 8 pages