Subracks and second homology of the conjugacy classes of finite projective special linear groups of degree two

TL;DR

This paper analyzes the structure of subracks within conjugacy classes of the group PSL(2,q), determines minimal non-abelian subracks, and relates these to the second homology and Schur multiplier of the group.

Contribution

It provides explicit descriptions of associated groups and second homology for conjugacy classes of PSL(2,q), extending understanding of their algebraic structure.

Findings

Explicit descriptions of associated groups for conjugacy classes

Classification of minimal non-abelian subracks

Connections between subrack structure and second homology

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 $q>3$.