Groups and quandles

TL;DR

This paper explores the properties of quandles and their enveloping groups, establishing isomorphisms, representations, and cohomological structures, with implications for finite and divisible groups.

Contribution

It provides new results on the structure, representations, and cohomology of quandles and their enveloping groups, including classifications and bounds for finite quandles.

Findings

Finite quandle enveloping groups admit faithful linear representations.

Finite injective quandles are subquandles of finite groups.

Finite subquandles of divisible groups are trivial quandles.

Abstract

We are intereseted in quandles and their enveloping groups. Various results are proven. We show that a quandle $Q$ and its image in the enveloping group $G(Q)$ have isomorphic enveloping groups. The image quandle is injective. For $Q$ a finite quandle, we show that $G(Q)$ admits a faithfull representation $\rho : G(Q)\to GL_n(\mathbb{Z})$ for some $n$; an irreducible representation of $G(Q)$ over $\mathbb{C}$ is finite dimensional an its degree divides the order of the group $Inn(Q)$ of inner automorphism of $Q$ and is bounded by $\sqrt{\vert Inn(Q) \vert }$. We determine the Malcev Lie algebra and the rational cohomology ring of $G(Q)$ for $Q$ finite. We prove that a finite injective quandle is a subquandle (for conjugacy) of a finite group. We also prove that the only finite subquandles (for conjugacy) of uniquely divisible groups are trivial quandles and that morphisms from quandles to nilpotent groups (for conjugacy) are constant on the indecomposable components. Implication of these results are considered.