Large totally symmetric sets

TL;DR

This paper investigates the size limitations of totally symmetric sets within groups, establishing a lower bound on group order based on the size of such sets and characterizing the largest possible sets.

Contribution

It proves that groups containing a totally symmetric set of size k must have order at least (k+1)! and characterizes the largest such sets in symmetric groups.

Findings

Groups with a totally symmetric set of size k have order ≥ (k+1)!

The set of transpositions {(1 i) | i=2,...,n} is essentially the largest totally symmetric set in S_n

Two exceptions exist where other sets achieve this bound

Abstract

A totally symmetric set is a subset of a group such that every permutation of the subset can be realized by conjugation in the group. The (non-)existence of large totally symmetric sets obstruct homomorphisms, so bounds on the sizes of totally symmetric sets are of particular use. In this paper, we prove that if a group has a totally symmetric set of size $k$, it must have order at least $(k+1)!$. We also show that with three exceptions, $\{(1 \; i)\mid i = 2,\ldots,n\} \subset S_n$ is the only totally symmetric set making this bound sharp; it is thus the largest totally symmetric set relative to the size of the ambient group.