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

Journal of Group Theory, vol. 0

Abstract

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.

Authors 4

1. Tim Boykett

   Johannes Kepler University of Linz · University of Applied Arts Vienna

   Affiliation as printed

   Time’s Up Research , Industriezeile 33b , Linz , 4020 Austria ; and Institute for Algebra, Johannes Kepler University Linz, 4040 Austria; and Design Investigations, University of Applied Arts, Vienna

   Design Investigations, University of Applied Arts, Vienna

2. Alice Devillers

   The University of Western Australia

   Affiliation as printed

   Centre for Mathematics of Symmetry and Computation , University of Western Australia , 35 Stirling Highway , Perth 6009 , Australia

3. Johannes Flake Aachen Algebra and Representation Theory

   RWTH Aachen University

   Affiliation as printed

   Algebra and Representation Theory , RWTH Aachen University , Pontdriesch 10-16, 52062 Aachen , Germany

4. S. P. Glasby corresponding

   The University of Western Australia

   Affiliation as printed

   Centre for Mathematics of Symmetry and Computation , University of Western Australia , 35 Stirling Highway , Perth 6009 , Australia