On the Classification of Perfect Prishchepov Groups
Layla Sorkatti, Ihechukwu Chinyere
TL;DR
The paper addresses the problem of classifying when Prishchepov cyclically presented groups $P(r,n,k,s,q)$ are perfect, focusing on the case $\gcd(n,6)=1$. The authors develop a framework based on the abelianization, represented by a circulant $n\times n$ matrix, and leverage Newton–Girard identities, Odoni’s unit criterion, and cyclotomic-unit arguments to translate perfectness into explicit congruence conditions. They prove that, under $\gcd(n,6)=1$, a perfect group is either of type $\tilde{\mathfrak{Z}}$ or satisfies one of the obvious congruences $k \equiv 1 \pmod{n}$, $k \equiv 1+q \pmod{n}$, $r \equiv 0 \pmod{n}$, or $s \equiv 0 \pmod{n}$; moreover, in the coprime-to-6 regime they derive a concrete corollary: $s=r-1$, $\gcd(n,q)=1$, and $\gcd(k-1-qr,n)=1$. This yields a complete classification of perfect Prishchepov groups under $\gcd(n,6)=1$ and connects to existing triviality results, with further avenues suggested for the case $\gcd(n,6)\neq 1$.
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.