Knapsack problem for nilpotent groups

TL;DR

This paper proves the undecidability of the Knapsack problem for certain nilpotent groups of class two and extends this result to broader classes like polycyclic and linear groups, highlighting computational limits in group theory.

Contribution

It establishes the undecidability of KP for nilpotent groups of class two with specific generator conditions and extends this to other complex group classes.

Findings

KP is undecidable for nilpotent groups of class two with ≥322 generators in the derived subgroup

Undecidability extends to certain polycyclic, linear, and higher nilpotency class groups

Results highlight computational boundaries in algebraic group problems

Abstract

We prove that Knapsack problem (KP) is undecidable for any group of nilpotency class two if the number of generators (without torsion) of the derived subgroup is at least 322. This result together with the fact that if KP is undecidable for a subgroup then it undecidable for the whole group allows us extend our result to certain classes of polycyclic groups, linear groups and nilpotent groups of higher nilpotency class.