Elementary detection at a fixed element

Put $A=\mathbb{Z}[{\zeta }_{\mathrm{\mid }G\mathrm{\mid }}]$. For $r\in R$, choose a Sylow $p$-subgroup $P_r$ of $C_G(r)$ and set $H_r=⟨r⟩\times P_r$. On $⟨r⟩$ the delta function ${\chi }_r(c)=\mathrm{\mid }⟨r⟩\mathrm{\mid }{\delta }_{c,r}$ lies in $AR(⟨r⟩)$ by Fourier inversion on this cyclic group. Inflate it across $P_r$ to ${\psi }_r\in AR(H_r)$ and form the following sum.

This lies in the $A$-scalar extension of $I_{{\mathcal{E}}_p}(G)$. Each ${\psi }_r$ is integer-valued, so the induction formula makes every value of $\psi$ rational. On the other hand $\psi$ is an $A$-linear combination of characters, hence all its values are algebraic integers. A rational algebraic integer is an integer, so $\psi$ is integer-valued.

If $r_0\in R$, a conjugate of $r_0$ lying in $H_r$ lies in $⟨r⟩$. The definition of ${\chi }_r$ and [F1] therefore give the following value.

Thus $\psi (r_0)$ is an integer prime to $p$. For arbitrary $g$, its $p^{\mathrm{\prime }}$-part is conjugate to some $r_0\in R$, so [F2] shows that $\psi (g)\equiv \psi (r_0)\equiv ̸0(\mathrm{m}\mathrm{o}\mathrm{d}p)$.

To use the integer-valued function from step 4.1, first let $\beta$ be any integer-valued class function on $G$. For each conjugacy-class representative $g$, put $C_g=⟨g⟩$ and define ${\theta }_g(c)=\mathrm{\mid }{C}_g\mathrm{\mid }{\delta }_{c,g}$ on $C_g$. The same cyclic Fourier inversion used in step 1.1 makes ${\theta }_g\in AR(C_g)$. By [F1], ${\mathrm{Ind}}_{C_g}^G{\theta }_g$ is $\mathrm{\mid }{C}_G(g)\mathrm{\mid }$ on the conjugacy class of $g$ and zero elsewhere. Consequently $\mathrm{\mid }G\mathrm{\mid }\beta ={\sum }_{[g]}\beta (g)[G:C_G(g)]{\mathrm{Ind}}_{C_g}^G{\theta }_g\mathrm{.}$ Every cyclic subgroup is $p$-elementary by splitting its order into coprime $p$-power and $p^{\mathrm{\prime }}$ factors, so this identity puts $\mathrm{\mid }G\mathrm{\mid }\beta$ in the $A$-scalar extension of $I_{{\mathcal{E}}_p}(G)$.

Assume $n\ge 1$ and put $e=p^{n-1}(p-1)$. Euler's congruence gives $\psi (g{)}^e\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}{p}^n)$ for every $g$. Thus $\beta =l({\psi }^e-1_G)\mathrm{/}\mathrm{\mid }G\mathrm{\mid }$ is an integer-valued class function, and step 5.1 puts $l({\psi }^e-1_G)$ in the $A$-scalar extension of $I_{{\mathcal{E}}_p}(G)$. The same is true of $l{\psi }^e$ by [F3] and step 1.1. Subtraction puts $l{1}_G$ in that scalar extension. If $n=0$, then $l=\mathrm{\mid }G\mathrm{\mid }$, so step 5.1 applied to $\beta =1_G$ gives the same conclusion directly.

The cyclotomic ring $A$ is a finite free $\mathbb{Z}$-module with the power basis $1,{\zeta }_{\mathrm{\mid }G\mathrm{\mid }},\dots ,{\zeta }_{\mathrm{\mid }G\mathrm{\mid }}^{\phi (\mathrm{\mid }G\mathrm{\mid })-1}$. Expand the relation from step 6.1 in this basis and take its coefficient of $1$. Since all inducing characters there lie in integral character rings, this yields $l{1}_G\in I_{{\mathcal{E}}_p}(G)$, including the case $n=0$.