TY - JOUR

T1 - The groups G satisfying a functional equation f(xk) = xf (x) for some k ∈ G

AU - Bernhardt, Dominik

AU - Boykett, Tim

AU - Devillers, Alice

AU - Flake, Johannes

AU - Glasby, Stephen P.

PY - 2022/11/1

Y1 - 2022/11/1

N2 - We study the groups G with the curious property that there exists an element k ∈ G and a function f: G → G such that f (xk) = xf (x) holds for all x ∈ 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. 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.

AB - We study the groups G with the curious property that there exists an element k ∈ G and a function f: G → G such that f (xk) = xf (x) holds for all x ∈ 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. 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.

UR - http://www.scopus.com/inward/record.url?scp=85131315303&partnerID=8YFLogxK

U2 - 10.1515/jgth-2021-0158

DO - 10.1515/jgth-2021-0158

M3 - Article

AN - SCOPUS:85131315303

SN - 1433-5883

VL - 25

SP - 1055

EP - 1081

JO - Journal of Group Theory

JF - Journal of Group Theory

IS - 6

ER -