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)| i = 2,...,n} ∈ S_n is the only totally symmetric set making this bound sharp, a fact which gives a new perspective on the automorphism group of S_n.

The author would like to thank Dan Margalit and Dan Minahan for their suggestion to push Theorem 1.1 farther than a simple bound. He is also grateful to Dan Margalit and an anonymous referee for their helpful comments.