General Results of Group Theory

Abstract

The ideal result of ‘abstract’ group theory would be a description of all possible groups up to isomorphism, completely independently of concrete realisations of the groups. In this generality, the problem is of course entirely impracticable. More concretely one could envisage the problem (which is still very wide) of classifying all finite groups. Since only a finite number of Cayley tables (multiplication tables) can be made up from a finite number of elements, there are only finitely many nonisomorphic groups of a given order; ideally one would like a rule specifying all finite groups of given order. For fairly small orders this can be done without much difficulty, and we run through the groups which arise.