Groups, conjugation and powers
Markus Szymik, Torstein Vik
TL;DR
The paper introduces power quandles as an enrichment of plain quandles that retains both conjugation and all power maps, yielding a stronger invariant for groups via the underlying structure $\mathrm{Pq}(G)$. It proves that $\mathrm{Pq}(G)$ determines central-quotient information $G/\mathrm{Z}(G)$ and, for finite groups, the center $\mathrm{Z}(G)$, embedding the problem in Baer’s central-quotient framework. A left adjoint $\mathrm{Gr}$ to the forgetful functor is constructed, showing $\mathrm{Gr}(\mathrm{Pq}(G))$ is a central extension of $G$ with abelian kernel $A(G)$, and giving a universal property that characterizes this extension; $A(G)$ is computed via a homological exact sequence involving $\mathrm{H}_2(G;\mathbb{Z})$ and a conjugacy-class based group $B(G)$. The essential-image analysis identifies pq-generated groups and shows that, for finite pq-generated groups, isomorphism of underlying power quandles implies isomorphism of the groups themselves after finite cancellations, linking to a broader question of whether all finite groups are pq-generated and how this interplays with the forgetful problem.
Abstract
We introduce the notion of the power quandle of a group, an algebraic structure that forgets the multiplication but keeps the conjugation and the power maps. Compared with plain quandles, power quandles are much better invariants of groups. We show that they determine the central quotient of any group and the center of any finite group. Any group can be canonically approximated by the associated group of its power quandle, which we show to be a central extension, with a universal property, and a computable kernel. This allows us to present any group as a quotient of a group with a power-conjugation presentation by an abelian subgroup that is determined by the power quandle and low-dimensional homological invariants.