For the second generation assertion use the permitted <Olshanskii centralizer lemma>, in the precise form
$$
C_{\mathbb C S_r}(\mathbb C S_{r-1})=\operatorname{alg}(Z_{r-1},X_r).
$$
In particular $Z_r\subseteq\operatorname{alg}(Z_{r-1},X_r)$. Induction puts every $Z_r$ in $\operatorname{alg}(1,X_1,\ldots,X_r)$. Conversely, if $A_r$ is the sum of all <transpositions> in $S_r$, then $A_r\in Z_r$ and $X_r=A_r-A_{r-1}\in GZ_n$. Therefore
$$
\boxed{GZ_n=\operatorname{alg}(1,X_1,\ldots,X_n).}
$$
Here generation is unital; the zero generator $X_1$ causes no problem.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-103/1/b/iv/solution