Equation satisfiability in solvable groups

TL;DR

This paper investigates the complexity of solving equations in finite groups, revealing that polynomial-time solvability implies the existence of a specific normal subgroup unless a major complexity hypothesis fails.

Contribution

It proves that finite groups with polynomial-time equation satisfiability necessarily have a nilpotent normal subgroup, linking group structure to computational complexity.

Findings

Polynomial-time solvability implies existence of a nilpotent normal subgroup.

The result holds unless the Exponential Time Hypothesis fails.

Provides a structural characterization of groups with efficiently solvable equations.

Abstract

The study of the complexity of the equation satisfiability problem in finite groups had been initiated by Goldmann and Russell (2002) where they showed that this problem is in polynomial time for nilpotent groups while it is NP-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups $G$ having a nilpotent normal subgroup $H$ with nilpotent factor $G/H$. This paper shows that such normal subgroup must exist in each finite group with equation satisfiability solvable in polynomial time, unless the Exponential Time Hypothesis fails.