Simple group
Despite the name, simple groups are far from "simple". The finite simple groups are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the way prime numbers are the basic building blocks of the integers. This is expressed by the Jordan-Hölder theorem. In a huge collaborative effort, the classification of finite simple groups was accomplished in 1982.
The only abelian simple groups are the cyclic groups of prime order. The famous theorem of Feit and Thompson states that every group of odd order is solvable. Therefore every finite simple group has even order unless it is cyclic of prime order.
Simple groups of infinite order also exist: simple Lie groups and the infinite Thompson groups T and V are examples of these.
