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

TL;DR

This paper characterizes groups called J-groups that admit a specific functional equation involving an element and a function, revealing their structure and conditions under which various classes of finite groups satisfy this property.

Contribution

It introduces the concept of J-groups, explores their properties, and provides criteria for finite nilpotent, p-groups, and regular p-groups to be J-groups.

Findings

Finite J-groups have odd order and are solvable.

Finite nilpotent groups of odd order with nilpotency class ≤6 are J-groups.

Certain p-groups with p>2 and specific size conditions are J-groups.

Abstract

We study the groups $G$ with the curious property that there exists an element $k\in G$ and a function $f\colon G\to G$ such that $f(xk)=xf(x)$ holds for all $x\in G$. 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 $c\le6$. If $G$ is a finite $p$-group, with $p>2$ and $p^2>2c-1$, 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 $m\geq1$ the subset of $p^m$-th powers is a subgroup), then we prove that $G$ is a $J$-group.