Functions On Finite Groups. Compatibility vs. Polynomiality

In the late sixties the concept of polynomial functions - mainly used in the context of fields up to then - has been transferred to arbitrary algebras. In this thesis we describe the polynomial functions on another class of finite groups, on nilpotent groups of class 2, ‘almost Abelian’ groups. Together with this description we get a fast method for counting polynomial functions on such groups, and a method for testing, whether and how a given function may be written as a polynomial function. In the second part of this thesis we discuss compatible functions on groups. A function from a group to itself is called compatible, if the elements of the same coset of any normal subgroup of the group are mapped into the same coset of this normal subgroup. Together with the older results about compatible functions on abelian groups and groups with a unique minimal normal subgroup we present results about groups with distributive minimal normal subgroups (a generalization) and direct products. Once more, we find efficient methods for the computation of such functions. In the third part we deal with the groups, where the concepts of polynomial and compatible functions coincide, 1-affine complete groups. Several new classes of such groups are presented.