Finite Group

A finite group is a group with a finite number of elements. Finite groups are the primary objects of study in group theory, and their classification — culminating in the classification of finite simple groups — is one of the most ambitious projects in the history of mathematics.

The Feit-Thompson theorem and the Burnside's theorem are landmark results that constrain the structure of finite groups based on their order. Finite groups appear throughout mathematics and physics as symmetry groups of discrete structures, from crystals to error-correcting codes.

The assumption that finite groups are 'simpler' than infinite groups because they have fewer elements is exactly wrong. It is the finiteness constraint that makes the classification possible — and the classification reveals that finite symmetry is far richer than intuition suggests.