Conjugation groups and structure groups of quandles

TL;DR

This paper investigates the structure of groups derived from quandles, especially those from conjugation groups, revealing their properties and computing the second quandle homology for symmetric groups.

Contribution

It introduces a new understanding of the groups s(onj(G)) for -groups, showing their embedding into G mp; bZ^m and computing the second quandle homology for symmetric groups.

Findings

s(onj(G)) injects into G mp; bZ^m for -groups.

The second quandle homology of onj(S_n) has rich torsion.

Properties of s(onj(G)) related to torsion, center, and derived subgroup are characterized.

Abstract

Quandles are certain algebraic structures showing up in different mathematical contexts. A group $G$ with the conjugation operation forms a quandle, $\operatorname{Conj}(G)$. In the opposite direction, one can construct a group $\operatorname{As}(Q)$ starting from any quandle $Q$. These groups are useful in practice, but hard to compute. We explore the group $\operatorname{As}(\operatorname{Conj}(G))$ for so-called $\overline{C}$-groups $G$. These are groups admitting a presentation with only conjugation and power relations. Symmetric groups $S_n$ are typical examples. We show that for $\overline{C}$-groups, $\operatorname{As}(\operatorname{Conj}(G))$ injects into $G \times \mathbb{Z}^m$, where $m$ is the number of conjugacy classes of $G$. From this we deduce information about the torsion, center, and derived group of $\operatorname{As}(\operatorname{Conj}(G))$. As an application, we compute the second quandle homology group of $\operatorname{Conj}(S_n)$ for all $n$, and unveil rich torsion therein.