• Corpus ID: 244270075

Groups, conjugation and powers

  title={Groups, conjugation and powers},
  author={Markus Szymik and Torstein Vik},
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… 



Permutations, power operations, and the center of the category of racks

ABSTRACT Racks and quandles are rich algebraic structures that are strong enough to classify knots. Here we develop several fundamental categorical aspects of the theories of racks and quandles and

A classifying invariant of knots, the knot quandle

Adams operations and symmetries of representation categories

Adams operations are the natural transformations of the representation ring functor on the category of finite groups, and they are one way to describe the usual lambda-ring structure on these rings.


A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a

Quandle cohomology is a Quillen cohomology

  • Markus Szymik
  • Mathematics
    Transactions of the American Mathematical Society
  • 2018
We show that the cohomology groups usually associated with racks and quandles agree with the Quillen cohomology groups for the algebraic theories of racks and quandles, respectively. We also explain

Groups with preassigned central and central quotient group

Every group G determines two important structural invariants, namely its central C(G) and its central quotient group Q(G) =G/C(G). Concerning these two invariants the following two problems seem to

Quandles: An Introduction to the Algebra of Knots

Knots and links Algebraic structures Quandles Quandles and groups Generalizations of quandles Enhancements Generalizd knots and links Bibliography Index


A sequence of new knot invariants is constructed by using the relationship between the theory of distributive groupoids and knot theory.Bibliography: 3 titles.

Cohomology of groups

  • L. Evens
  • Engineering
    Oxford mathematical monographs
  • 1991
A rink-type roller skate is provided with a plastic sole plate. To mount a toe stop on the skate, a novel bushing is embedded in the sole plate. The bushing has relatively small diameter ends and a

Alexander–Beck modules detect the unknot

We introduce the Alexander-Beck module of a knot as a canonical refinement of the classical Alexander module, and we prove that this new invariant is an unknot-detector.