Indecomposable racks of order p^2

TL;DR

This paper classifies indecomposable racks of order p^2, revealing their structure, isomorphism classes, and properties of associated quandles, including their affine nature and cohomology groups.

Contribution

It provides a complete classification of indecomposable racks of order p^2 and proves that such quandles are affine, also computing their second cohomology groups.

Findings

Number of isomorphism classes: 2p^2 - 2p - 2

Indecomposable quandles of order p^2 are affine

Second cohomology groups are trivial for these quandles

Abstract

We classify indecomposable racks of order p^2 (p a prime). There are 2p^2 - 2p - 2 isomorphism classes, among which 2p^2 - 3p - 1 correspond to quandles. In particular, we prove that an indecomposable quandle of order p^2 is affine (=Alexander). One of the results yielding this classification is the computation of the quandle nonabelian second cohomology group of an indecomposable quandle of prime order; which turns out to be trivial, as in the abelian case.