Permutation groups generated by a \((2,2)\)-cycle and an \(n\)-cycle (Q1392335)

Let \(a\) be an \(n\)-cycle in the symmetric group \(S_n\) and let \(b\) be a double-transposition (a \((2,2)\)-cycle) and set \(H:=\langle a,b\rangle\). It is proved in the Theorem of the paper that \(H\) is full (i. e. symmetric or alternating) or \(n=6\) and \(H\cong S_5\) or \(n=7\) and \(H\cong\text{PSL}_2(7)\). The proofs are elementary in graph-theoretic spirit. Applications of the result to monodromy groups of planar trees and to Galois theory over the rationals are given.