An equivariant version of an index formula of Fröhlich and McCulloh (Q1267289)

By Iwasawa's index formula, the order of the minus class group \(C \ell (\mathbb{Z}[\xi_p])^-\) equals the index of \(J_1^-\) in \(\mathbb{Z} G^-_1\), where \(G_1= \text{Gal}(\mathbb{Q} (\xi_p)/ \mathbb{Q})\) and \(J_1\) is the Stickelberger ideal in \(\mathbb{Z} G_1\). Using this and an index calculation of Fröhlich, \textit{L. McCulloh} [J. Algebra 68, 443-452 (1981; Zbl 0472.12004)] has shown that, for any abelian elementary \(p\)-group \(\Gamma\), the order of \(C\ell (\mathbb{Z} \Gamma)^-\) equals the index of \(J^-\) in \(\mathbb{Z} G^-\), where \(G\) is a suitable subgroup of \(\Aut_\mathbb{Z} (\Gamma)\) and \(J\) a straightforward generalization of the Stickelberger ideal. Now the Main Conjecture yields an equivalent version of Iwasawa's index formula, namely that \(C\ell(\mathbb{Z}[\xi_p])^-\) and \(\mathbb{Z} G^-_1/J_1^-\) have isomorphic composition series as \(\mathbb{Z} G_1\)-modules. Using this, the author proves that \(C\ell (\mathbb{Z} \Gamma)^-\) and \(\mathbb{Z} G^-/J^-\) have isomorphic \(\mathbb{Z} G\)-composition series.