String C-groups as transitive subgroups of Sym(n)

TL;DR

This paper investigates string C-groups that are transitive subgroups of Sym(n), establishing an upper bound on their rank and classifying exceptions, with a conjecture that only Sym(n) itself needs exclusion.

Contribution

The paper proves an upper bound on the rank of certain string C-groups and classifies the finite exceptions, advancing understanding of their structure within symmetric groups.

Findings

Rank of such string C-groups is at most n/2+1

Finite exceptions to the bound are classified

Conjecture that only Sym(n) needs to be excluded

Abstract

If $\Gamma$ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group Sym(n) (other than Sym(n) and the alternating group Alt(n)), then the rank of $\Gamma$ is at most $n/2+1$, with finitely many exceptions (which are classified). It is conjectured that only the symmetric group has to be excluded.