Subrack Lattices of Conjugation Racks

Abstract

A rack is a set with a binary operation such that left multiplications are automorphisms of the set and a quandle is a rack satisfying a certain condition. Let $S$ be a subset of a finite group $G$ which is closed under the conjugation operation $a\triangleright b := aba^{-1}$. The set $S$ with the conjugation operation $\triangleright$ is a quandle. We call those objects \emph{conjugation racks}. The prime examples are \begin{itemize} \item the group rack $(G,\triangleright)$, \item the conjugacy class rack $(C,\triangleright)$, where $C$ is a conjugacy class in $G$, and \item the $p$-power rack $(G_p,\triangleright)$, where $p$ is a prime and $G_p$ is the set of all elements in $G$ whose order is a power of $p$. \end{itemize} The set of all subracks of a finite rack form a lattice under inclusion. In this paper we study the subrack lattices of the conjugation racks. In particular, we show that the subrack lattice can be associated with a subposet of a partition lattice as well as with a subposet of an integer partition lattice in a canonical way if the rack is connected. And, if the rack is not connected, the study of the homotopy properties of the subrack lattice can be reduced into the study of the homotopy properties of the subposet of parabolic subracks. We also prove that for a certain class of $p$-power racks the order of a Sylow $p$-subgroup divides the reduced Euler characteristic of the subrack lattice of the $p$-power rack. This statement can be considered as the rack analogue of a result by Brown in the field of subgroup complexes regarding the Euler characteristic of the poset of nontrivial $p$-subgroups of a group.