Annihilation of torsion in homology of finite $m$-AQ quandles

TL;DR

This paper generalizes the classical result that the order of a finite group annihilates its homology to a broader class of algebraic structures called $m$-AQ quandles, showing similar torsion annihilation properties.

Contribution

It introduces $m$-AQ quandles, a new generalization of quasigroup quandles, and investigates torsion annihilation in their homology groups.

Findings

Torsion in rack and quandle homology of finite $m$-AQ quandles is annihilated by their order.

The classical torsion annihilation result extends to $m$-AQ quandles.

Connected quandles do not necessarily exhibit torsion annihilation.

Abstract

It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. Similarly, we have proved that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. However, it does not hold for connected quandles in general. In this paper, we define an $m$-almost quasigroup ($m$-AQ) quandle which is a generalization of a quasigroup quandle and study annihilation of torsion in its rack and quandle homology groups.