Large totally symmetric sets
Noah Caplinger

TL;DR
This paper investigates the size limitations of totally symmetric sets within groups, establishing a lower bound on group order based on the size of such sets and characterizing the largest possible sets.
Contribution
It proves that groups containing a totally symmetric set of size k must have order at least (k+1)! and characterizes the largest such sets in symmetric groups.
Findings
Groups with a totally symmetric set of size k have order ≥ (k+1)!
The set of transpositions {(1 i) | i=2,...,n} is essentially the largest totally symmetric set in S_n
Two exceptions exist where other sets achieve this bound
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 , it must have order at least . We also show that with three exceptions, 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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Taxonomy
TopicsLimits and Structures in Graph Theory · Finite Group Theory Research · Graph theory and applications
