On the lattice of subracks of the rack of a finite group

Istvan Heckenberger, John Shareshian, Volkmar Welker

Published 2015-12-04Version 1

In this paper we initiate the study of racks from the combined perspective of combinatorics and finite group theory. A rack R is a set with a self-distributive binary operation. We study the combinatorics of the partially ordered set {\cal R}(R) of all subracks of R with inclusion as the order relation. Groups G with the conjugation operation provide an important class of racks. For the case R = G we show that -> the order complex of {\cal R}(R) has the homotopy type of a sphere, -> the isomorphism type of {\cal R}(R) determines if G is abelian, nilpotent, supersolvable, solvable or simple, -> {\cal R}(R) is graded if and only if G is abelian, G = S_3, G = D_8 or G = Q_8. In addition, we provide some examples of subracks R of a group G for which {\cal R}(R) relates to well studied combinatorial structures. In particular, the examples show that the order complex of {\cal R}(R) for general R is more complicated than in the case R = G.