Large totally symmetric sets

2022/08/18 by Caplinger, Noah
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2208.09050

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)| i = 2,…,n\ ⊂ 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.

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