Subracks and second homology of the conjugacy classes of finite projective special linear groups of degree two
Istvan Heckenberger, Fengchang Li
TL;DR
The paper provides a detailed subrack analysis for the conjugacy classes of PSL$(2,q)$ with $q>3$ by leveraging Dickson's subgroup classification, yielding a complete description of subracks and minimal non-abelian racks. It introduces a general framework linking the associated group $ ext{As}(X)$ of a generating conjugacy class to the Schur multiplier, and then applies this to PSL$(2,q)$ to obtain explicit forms for $ ext{As}(X)$ and the second quandle homology $H_2(X)$ across all non-trivial conjugacy classes, including the exceptional case $q=9$. The results illuminate the interplay between rack theory and finite group structure, with implications for Nichols algebras and braided Hopf algebras built from these racks. Overall, the work provides a comprehensive toolkit for understanding the generation, embedding, and cohomological properties of conjugacy-classes racks in classical groups, particularly PSL$(2,q)$.
Abstract
We describe the subracks of the conjugacy classes of $\mathrm{PSL}(2,q)$ based on Dickson's theorem on subgroups of $\mathrm{PSL}(2,q)$. All minimal non-abelian subracks of $\mathrm{PSL}(2,q)$ are determined. Further, we provide a general result on the relationship of associated groups of conjugacy classes in perfect groups to the Schur multiplier of the group. This allows us to conclude explicit descriptions of the associated groups and the second homology of the conjugacy classes of $\mathrm{PSL}(2,q)$ for $q>3$.