ABSTRACT
9.1 Few concepts have penetrated mathematics so deeply as the concept of group and group action. In the section on Historical Comments we include a brief history of the role of groups in the solution of equations e.g. in Galois Theory. Equally well known is A. Klein's “Erlangen Programm” based on the principle that most of geometry can be reduced to problems concerning invariants of group actions. The intuitive side of abstract groups is that they still can be viewed as transformation groups of something, even if this is a completely abstract object like a field or even the group itself. Faithful to the principles of category theory we present the object together with its morphisms, that is group homomorphisms.
