# Detecting laws in power subgroups

## TL;DR

This paper investigates which group laws can be determined by their behavior in power subgroups, proving detectability for certain laws like nilpotency and Engel laws, but showing failure for solvability, and analyzing the complexity of detecting commutativity.

## Contribution

It establishes that nilpotency and low-order Engel laws are detectable in power subgroups, introduces a counterexample for solvability, and studies the complexity of detecting commutativity.

## Key findings

- Nilpotency of class at most c is detectable in power subgroups.
- The k-Engel law for k ≤ 4 is detectable in power subgroups.
- Detectability fails for solvability with certain derived lengths.

## Abstract

A group law is said to be detectable in power subgroups if, for all coprime $m$ and $n$, a group $G$ satisfies the law if and only if the power subgroups $G^m$ and $G^n$ both satisfy the law. We prove that for all positive integers $c$, nilpotency of class at most $c$ is detectable in power subgroups, as is the $k$-Engel law for $k$ at most 4. In contrast, detectability in power subgroups fails for solvability of given derived length: we construct a finite group $W$ such that $W^2$ and $W^3$ are metabelian but $W$ has derived length $3$. We analyse the complexity of the detectability of commutativity in power subgroups, in terms of finite presentations that encode a proof of the result.