The pseudovariety of all nilpotent groups is tame

TL;DR

This paper proves that the class of all finite nilpotent groups is tame in the context of pseudovarieties, extending previous results on finite p-groups, but also shows it is not completely tame.

Contribution

It generalizes the tameness property from finite p-groups to all finite nilpotent groups, highlighting the broader applicability and limitations.

Findings

Finite nilpotent groups form a tame pseudovariety.

The pseudovariety of finite nilpotent groups is not completely tame.

Extension of tameness results from p-groups to nilpotent groups.

Abstract

It has been shown that for every prime number $p$, the pseudovariety $\mathsf{G}_p$ of all finite $p$-groups is tame with respect to an implicit signature containing the canonical implicit signature. In this paper, we generalize this result and we show that the pseudovariety of all finite nilpotent groups is tame but it is not completely tame.