Computable two-variable laws for finite groups with abelian Sylow subgroups

Abstract

For each prime $p$ we construct a computable sequence $a_{p,n}$ of words in two variables such that, for every finite group $G$ and every $n\geq |G|$, the identity $a_{p,n}=1$ holds in $G$ if and only if the Sylow $p$-subgroups of $G$ are abelian. When this condition fails, one fixed pair in $G$ makes every word in that tail nontrivial. The construction uses a decidable finite test and a terminating search through word pairs. It concerns the explicit-sequence question in Kourovka Problem 11.15; the notion of explicitness established here is total computability by finite-group search.