A class-field theoretical calculation (Q873843)

Let \(k\) be a function field of one variable over a finite field \(\mathbb{F}_p\), where \(p\) is a prime number. Let \(K/k\) be a finite abelian extension with Galois group \(\Gamma:=\text{Gal}(K/k)\). Denote by \(S\) the set of ramified primes of \(K\) in \(K\) over \(k\). Let \(K_S^{ab,p}\) be the maximal pro-\(p\) abelian extension of \(K\) unramified outside \(S\) with Galois group \(H:=\text{Gal}(K_S^{ab,p}/K)\). Then \(K_S^{ab,p}\) is Galois over \(k\) with Galois group \(G:=\text{Gal}(K_S^{ab,p}/k)\). Thus there is an exact sequence \[ 1\to H\to G\to \Gamma \to 1. \] The action of \(\Gamma\) on \(H\) is \(\gamma\circ h:=\tilde{\gamma}h\tilde{\gamma}^{-1},\) for all \(h\in H\) and \(\gamma\in \Gamma\), where \(\tilde{\gamma}\) is any lift of \(\gamma\) to \(G\). Consider the augmentation ideal \(I_{\Gamma}:=\langle\gamma-1|\gamma\in\Gamma\rangle\) and \(H\), we have \((\gamma-1)\circ h=\tilde{\gamma}h\tilde{\gamma}^{-1}h^{-1}=\) the commutator \([\tilde{\gamma},h]\) of \(\tilde{\gamma}\) and \(h\), for all \(\gamma\in\Gamma\) and \(h\in H\). As a result, we have an the inclusion \(I_{\Gamma}\circ H\subseteq [G,G]\), where \([G,G]\) is the commutator subgroup of \(G\). This paper gives a necessary and sufficient condition for the other inclusion \(I_{\Gamma}\circ H \supseteq [G,G]\) to hold, i.e. \(I_{\Gamma}\circ H =[G,G]\) if and only if the \(p\)-sylow subgroup \(\Gamma^{(p)}\) of \(\Gamma\) is cyclic. The author first gives a sufficient condition for \(I_{\Gamma}\circ H =[G,G]\) to hold, i.e. the condition \(H/I_{\Gamma}\circ H\) has no torsion; then using class field theory he shows an isomorphism \(T(H/I_{\Gamma}\circ H)\cong \wedge^2 \Gamma^{(p)}\), where \(T(H/I_{\Gamma}\circ H)\) is the torsion subgroup of \(H/I_{\Gamma}\circ H\); thus he concludes that if \(\Gamma^{(p)}\) is cyclic then \(I_{\Gamma}\circ H =[G,G]\); finally the author uses class field theory and a theorem of \textit{H. Kisilevsky} [J. Number Theory 44, No. 3, 352--355 (1993; Zbl 0780.11058)] to show that the condition \(\Gamma^{(p)}\) is cyclic is in fact necessary.