Chapter 4.4, Problem 31E

A subgroup $H$ of the group $S_n$ is called transitive on $B=\{1,2,\mathrm{....},n\}$ if for each pair $i,j$ of elements of $B$ there exists an element $h\in H$ such that $h(i)=j$. Suppose $G$ is a group that is transitive on $\{1,2,\mathrm{....},n\}$, and let $H_i$ be the subgroup of G that leaves $i$ fixed:

$H_i=\{g\in G|g(i)=i\}$

For $i=1,2,\mathrm{...},n$. Prove that $\left|G\right|=n·\left|H_i\right|$.