Groups, conjugation and powers

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.