Hyperelementary permutation subring reduction

By the projection formula, $({\mathrm{Ind}}_H^G{1}_H)({\mathrm{Ind}}_K^G{1}_K)={\mathrm{Ind}}_H^G({\mathrm{Res}}_H^G{\mathrm{Ind}}_K^G{1}_K)$. Mackey's formula expresses the restricted character as a finite sum of ${\mathrm{Ind}}_{H\cap xK{x}^{-1}}^H{1}_{H\cap xK{x}^{-1}}$. Each intersection is hyperelementary by subgroup closure; transitivity then makes every summand induced back to $G$ equal to ${\mathrm{Ind}}_{H\cap xK{x}^{-1}}^G{1}_{H\cap xK{x}^{-1}}$. Thus products of the displayed generators stay in $P({\mathcal{H}}^{\mathrm{\prime }})$, and bilinearity makes its additive span a subring.

If $1_H\in I_{\mathcal{E}}(H)$, express it by [F3] as an integral sum of characters induced from elementary subgroups of $H$. Inducing this sum to $G$ and using transitivity puts ${\mathrm{Ind}}_H^G{1}_H$ in $I_{\mathcal{E}}(G)$. Apply this to every summand of a relation for $1_G$ in $P({\mathcal{H}}^{\mathrm{\prime }})$. ∎