The problem of determining the number of polynomial functions on a given group can be attacked from various perspectives. One is to compute all these functions in an efficient way. This can be useful for small examples and helps testing and finding theoretical hypotheses.
An important starting point for theoretical research is the examination of the properties of the lattice of normal subgroups and their commutators. Here a lot of results from the area of universal algebra can be used successfully (cf. ). Another approach is a ``pointwise'' investigation of the group (i.e., playing with the elements of the group instead of its subgroups). Nilpotent groups and in particular p-groups are a point where these different approaches come together. Efficient algorithms are known for p-groups: structure theory, as well as an elementary (``pointwise'') analysis of p-groups are heavily investigated areas.