2015/12/04 by Heckenberger, Istvan, Shareshian, John, Welker, Volkmar · 1 citation
#05E45 #20D30 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.1512.01459
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 = S3, G = D8 or G = Q8. 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.