Some notes on extended equation solvability and identity checking for groups

TL;DR

This paper explores the computational complexity of solving equations and checking identities in extended algebraic structures derived from finite groups, providing a uniform extension method and characterizing specific group classes.

Contribution

It introduces a uniform term extension that induces NP-completeness and co-NP-completeness in solving equations and identity checking, respectively, for a broad class of groups.

Findings

Extending groups by a specific term operation increases complexity.

A uniform extension suffices for many solvable, non-nilpotent groups.

Characterization of groups where commutator extension is enough.

Abstract

Every finite non-nilpotent group can be extended by a term operation such that solving equations in the resulting algebra is NP-complete and checking identities is co-NP-complete. This result was firstly proven by Horv\'ath and Szab\'o; the term constructed in their proof depends on the underlying group. In this paper we provide a uniform term extension that induces hardness. In doing so we also characterize a big class of solvable, non-nilpotent groups for which extending by the commutator operation suffices.