Example 64.30.2. Fix an integer q\geq 1
\begin{matrix} 1
& \to
& G = \widehat{\mathbf{Z}}^{(q)}
& \to
& \Gamma
& \to
& \widehat{\mathbf{Z}}
& \to
& 1
\\ & & = \prod _{l\not\mid q} \mathbf{Z}_ l
& & F
& \mapsto
& 1
\end{matrix}
with FxF^{-1} = ux, u \in (\widehat{\mathbf{Z}}^{(q)})^*. Just using the trivial modules \mathbf{Z}/m\mathbf{Z} we see
q^ n - (qu)^ n \equiv \sum \nolimits _{d|n} d\# S_ d
in \mathbf{Z}/m\mathbf{Z} for all (m, q)=1 (up to u \to u^{-1}) this implies qu = a\in \mathbf{Z} and |a| < q. The special case a = 1 does occur with
\Gamma = \pi _1^ t(\mathbf{G}_{m, \mathbf{F}_ p}, \overline\eta ), \quad \# S_1 = q - 1, \quad \text{and}\quad \# S_2 = \frac{(q^2-1)-(q-1)}{2}
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