On the Classification of Perfect Prishchepov Groups

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 $r\ge s$) or $q(r+s)$. This settles the classification of perfect Prishchepov groups under the co-primality condition.