Simple polyadic groups

TL;DR

This paper classifies simple polyadic groups by linking them to ordinary groups and their automorphisms, providing criteria for their simpleness from algebraic and group-theoretic perspectives.

Contribution

It introduces two definitions of simplicity for polyadic groups and establishes necessary and sufficient conditions for each, connecting polyadic groups to classical group theory.

Findings

Criteria for UAS simpleness established

Criteria for GTS simpleness established

Classification of simple polyadic groups achieved

Abstract

The main aim of this article is to establish a classification of simple polyadic groups in terms of ordinary groups and their automorphisms. We give two different definitions of simpleness for polyadic groups, from the point of views of universal algebra, UAS (universal algebraically simpleness), and group theory, GTS (group theoretically simpleness). We obtain the necessary and sufficient conditions for a polyadic group to be UAS or GTS.