Large totally symmetric sets

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, $\{(1i)\mid i=2,\dots ,n\}\subset S_n$ 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.