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

TL;DR

This paper explores the combinatorial and algebraic structure of subracks within finite groups, revealing topological properties and characterizations of the underlying groups based on the lattice of subracks.

Contribution

It introduces the study of the lattice of subracks of a finite group and characterizes group properties through the structure of this lattice.

Findings

The order complex of the subrack lattice has the homotopy type of a sphere.

The lattice determines if the group is abelian, nilpotent, supersolvable, solvable, or simple.

The subrack lattice is graded only for specific groups like abelian groups, S_3, D_8, or Q_8.

Abstract

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.