Decidability of affine completeness for finite groups

Determine whether there exists an algorithm that, given a finite group G (viewed as an algebra with its group operations), decides whether G is affine complete, i.e., whether for every arity k each total congruence-preserving function f: G^k -> G is a polynomial (term) function of G.

Background

The paper discusses polynomial completeness properties such as affine completeness, where an algebra A is affine complete if every total congruence-preserving function on A is a polynomial function. In the context of groups, this question specializes to determining whether a given finite group is affine complete.

Although there are characterization results and studies of affine complete groups, the algorithmic decidability of testing affine completeness for a given finite group remains unresolved, as explicitly noted in the introduction.