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

Abstract:

In this paper we initiate the study of the poset of subsets $R$ of a finite group $G$ that are closed under conjugation by elements from $R$. We write $\mathcal {R}(G)$ for the set of those $R$ and order $\mathcal {R}(G)$ by inclusion. We show:

the isomorphism type of the partially ordered set $\mathcal {R}(G)$ determines if $G$ is abelian, nilpotent, supersolvable, solvable, or simple,

$\mathcal {R}(G)$ is graded if and only if $G$ is abelian, $G = S_3$, $G = D_8$, or $G = Q_8$,

the order complex of $\mathcal {R}(G)$ has the homotopy type of a sphere, and

$\mathcal {R}(G)$ is (sequentially) Cohen-Macaulay (resp. shellable) if and only if $G$ is abelian.

If the finite group $G$ is considered as a rack with conjugation as its rack operation, then $\mathcal {R}(G)$ can equivalently be defined as the partially ordered set of all subracks of $G$. For general (finite) racks $R$ the structure of the partially ordered set $\mathcal {R}(R)$ of its subracks remains mysterious. We provide some examples of subracks $R$ of a group $G$ for which $\mathcal {R}(R)$ relates to well studied combinatorial structures. In particular, the examples show that the order complex of $\mathcal {R}(R)$ for general $R$ is more complicated than in the case $R = G$.