Gyro-Groups, Gyro-splittings and Co-homology

TL;DR

This paper explores the structure of gyro-groups linked to groups and extensions, introduces new co-homology concepts, and discusses obstructions and invariants like gyro-Schur Multiplier and gyro-Milnor K2.

Contribution

It introduces the concepts of gyro-groups, gyro-Schur Multiplier, and gyro-Milnor K2, and analyzes their co-homological properties and obstructions.

Findings

Defined gyro-groups associated with groups and extensions

Described obstructions using co-homology

Introduced gyro-Schur Multiplier and gyro-Milnor K2 groups

Abstract

In this paper, we study gyro-groups associated to groups, group extensions admitting gyro-sections, and corresponding co-homologies. We also describe the obstructions in terms of co-homomology. The notion of gyro-Schur Multiplier and that of gyro-Milnor $K_{2}$ group are introduced.