Example 63.30.2. 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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