Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 103 / 1 / b / iv / Solution

For the second generation assertion use the permitted Olshanskii centralizer lemma, in the precise formIn particular $Z_r\subseteq \mathrm{alg}(Z_{r-1},X_r)$. Induction puts every $Z_r$ in $\mathrm{alg}(1,X_1,\dots ,X_r)$. Conversely, if $A_r$ is the sum of all transpositions in $S_r$, then $A_r\in Z_r$ and $X_r=A_r-A_{r-1}\in G{Z}_n$. ThereforeHere generation is unital; the zero generator $X_1$ causes no problem.