Upper Bounds for Totally Symmetric Sets

TL;DR

This paper classifies totally symmetric sets in various groups, establishes bounds on their sizes, and derives restrictions on homomorphisms, notably showing braid group homomorphisms to solvable groups have cyclic images.

Contribution

It provides full classifications and bounds for totally symmetric sets in specific groups, advancing understanding of group homomorphisms and their limitations.

Findings

Classified totally symmetric sets in certain groups

Bounded sizes of these sets in various groups

Proved homomorphisms from braid groups to solvable groups have cyclic images

Abstract

Totally symmetric sets are a recently introduced tool for studying homomorphisms between groups. In this paper, we give full classifications of totally symmetric sets in certain families of groups and bound their sizes in others. As a consequence, we derive restrictions on possible homomorphisms between these groups. One sample application of our results is that any homomorphism of a braid group to a direct product of solvable groups must have cyclic image.