About the length of laws for finite groups

TL;DR

This paper establishes improved upper bounds on the length of laws valid for all finite groups up to size n, utilizing the classification of finite simple groups and providing stronger bounds for nilpotent and solvable groups.

Contribution

It introduces new upper bounds for the length of laws for finite groups, leveraging classification techniques and extending results to specific group classes.

Findings

Improved bounds for laws in all finite groups up to size n

Stronger bounds for nilpotent and solvable groups

Utilization of classification of finite simple groups

Abstract

We prove new upper bounds for the length of laws that hold for all groups of size at most $n$ -- improving on previous results of Bou-Rabee and Kassabov-Matucci. The methods make use of the classification of finite simple groups. Stronger bounds are proved in case the groups are assumed to be nilpotent or solvable.