The groups G satisfying a functional equation f(xκ) = xf(x) for some κ ∈ G

We study the groups G with the curious property that there exists an element k 2 G and a function f W G ! G such that f .xk/ D xf .x/ holds for all x 2 G. This property arose from the study of near-rings and input-output automata on groups. We call a group with this property a J -group. Finite J -groups must have odd order, and hence are solvable. We prove that every finite nilpotent group of odd order is a J -group if its nilpotency class c satisfies c 6 6. If G is a finite p-group, with p > 2 and p2 > 2c � 1, then we prove that G is J -group. Finally, if p > 2 and G is a regular p-group or, more generally, a power-closed one (i.e., in each section and for each m > 1, the subset of pm-th powers is a subgroup), then we prove that G is a J -group.