ABSTRACT
The classification problem for finite groups is the question of how to list all finite groups “up to isomorphism,” i.e., one representative for each class of isomorphic groups. Having such a list, presumably we could verify as sertions about groups simply by going down the list. However, there are several difficulties with this approach. First of all, the list must be infinite, since (Z m, + ) is a finite group for each m. Moreover, even if we had a complete list of finite groups, it might be impossible in practice to verify a given assertion. For example, 1, 2 ,3 ,... is a list of the natural numbers, but we do not know if there are any odd perfect numbers. Finally, the list of finite groups might be less enlightening than the body of theorems used to obtain the list. Nevertheless, the classification problem is the focal research problem in the theory of finite groups, and the ensuing results have been of great use.
