 

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


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, a fact which gives a new perspective on the automorphism group of S_{n}.


Acknowledgements
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, 111832 E. 58th Street
Chicago, IL 60637, USA
ncaplinger@uchicago.edu

