New York Journal of Mathematics
Volume 29 (2023), 931-938


Noah Caplinger

Large totally symmetric sets

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Published: July 25, 2023.
Keywords: Totally symmetric set, Symmetric Group, Automorphism.
Subject [2020]: 20F99.

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} ∈ Sn is the only totally symmetric set making this bound sharp, a fact which gives a new perspective on the automorphism group of Sn.


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.

Author information

Noah Caplinger
University of Chicago
Department of Mathematics
Eckhart Hall, 1118-32 E. 58th Street
Chicago, IL 60637, USA