Binary transformation groups and topological fields

TL;DR

This paper introduces a duality between semitransitive binary group actions on topological spaces and topological fields with specific multiplicative groups, establishing new categorical equivalences and characterizations.

Contribution

It establishes a duality theorem linking semitransitive binary group actions to topological fields and characterizes groups that can be multiplicative groups of such fields.

Findings

Finite groups act semitransitively only on sets with prime power cardinality.

A complete characterization of groups as multiplicative groups of topological fields.

Duality theorem creates a categorical equivalence between certain group actions and topological fields.

Abstract

The notion of a semitransitive binary action of a group $G$ on a topological space is introduced. A duality theorem is proved, establishing a bijective correspondence between semitransitive distributive binary $G$-spaces and topological fields whose multiplicative group is isomorphic to $G$. This result yields an equivalence between the category of semitransitive distributive binary $G$-spaces and the category of topological fields with multiplicative group $G$. As applications of the duality theorem, two important results are established. It is shown that a finite group can act semitransitively, distributively, and binarily only on finite sets whose cardinality is a power of a prime number. A complete characterization of those groups that can appear as multiplicative groups of topological fields is also obtained.