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

Abstract We study the groups 𝐺 with the curious property that there exists an element k ∈ G k\in G and a function f : G → G f\colon G\to G such that f ⁢ ( x ⁢ k ) = x ⁢ f ⁢ ( x ) f(xk)=xf(x) holds for all x ∈ G x\in 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 c\leqslant 6 . If 𝐺 is a finite 𝑝-group, with p > 2 p>2 and p 2 > 2 ⁢ c - 1 p^{2}>2c-1 , then we prove that 𝐺 is 𝐽-group. Finally, if p > 2 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 m\geqslant 1 , the subset of p m p^{m} -th powers is a subgroup), then we prove that 𝐺 is a 𝐽-group.