On the Classification of Perfect Prishchepov Groups

TL;DR

This paper classifies when a family of cyclically presented groups called Prishchepov groups are perfect, providing a clear criterion based on divisibility conditions for the case when n is coprime to 6.

Contribution

It proves a conjecture that characterizes perfect Prishchepov groups under the co-primality condition, unifying many classical cases.

Findings

Provides a divisibility criterion for perfectness of Prishchepov groups

Settles the classification of perfect groups in this family for n coprime to 6

Unifies various classical cases under a single framework

Abstract

We study the Prishchepov groups $P(r,n,k,s,q)$, a unifying family of cyclically presented groups that encompasses many classical cases. For $n$ coprime to $6$, we prove a conjecture essentially characterizing when these groups are perfect: namely, $n$ divides either $2(k-1)-q$ (if $r \geq s$) or $q(r+s)$. This settles the classification of perfect Prishchepov groups under the co-primality condition.