On core quandles of groups

TL;DR

This paper explores the properties of core quandles derived from groups, examining their structure, embedding conditions, and generation bounds, thereby connecting group identities with involutory quandle properties.

Contribution

It provides new insights into the structure of core quandles, necessary conditions for their embeddability in groups, and bounds on their generation, advancing the understanding of involutory quandles.

Findings

Identified conditions for embedding involutory quandles into core quandles of groups.

Established bounds on the number of elements needed to generate core quandles.

Explored relationships between group identities and identities in their core quandles.

Abstract

We review the definition of a quandle, and in particular of the core quandle $\mathrm{Core}(G)$ of a group $G$, which consists of the underlying set of $G$, with the binary operation $x\lhd y = x y^{-1} x$. This is an involutory quandle, i.e., satisfies the identity $x\lhd (x\lhd y) = y$ in addition to the other identities defining a quandle. Trajectories $(x_i)_{i\in\mathbb{Z}}$ in groups and in involutory quandles (in the former context, sequences of the form $x_i = x z^i$ where $x,z\in G,$ among other characterizations; in the latter, sequences satisfying $x_{i+1}= x_i\lhd\,x_{i-1})$ are examined. A family of necessary conditions for an involutory quandle to be embeddable in the core quandle of a group is noted. Some implications are established between identities holding in groups and in their core quandles. Upper and lower bounds are obtained on the number of elements needed to generate the quandle $\mathrm{Core}(G)$ for $G$ a finitely generated group. Several questions are posed.