Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics I / 7 / a / iv / Solution

For distinct $i,j\ne 1$, direct conjugation givesThus the given star transpositions generate every transposition. Every cycle can in turn be expressed asBy the disjoint cycle decomposition, every permutation is a product of cycles, hence of transpositions, hence of the given star transpositions. ThereforeFor $n=1$, the empty generating set of a group generates the trivial group $S_1$.