The generation of the augmentation ideal in profinite groups.
Let \(G\) be a profinite group, let \(\widehat{\mathbb Z}G\) be the completed group algebra over \(\widehat{\mathbb Z}\) and let \(I_{\widehat{\mathbb Z}}G\) be its augmentation ideal. The author finds a formula for computing the minimum number of generators of \(I_{\widehat{\mathbb Z}}G\) as a \(\widehat{\mathbb Z}G\)-module, generalizing a result obtained in the finite case by \textit{J. Cossey, K. W. Gruenberg} and \textit{L. G. Kovács} [J. Algebra 28, 597-603 (1974; Zbl 0293.20033)]. As a consequence an example is produced of a profinite group \(G\) which is not finitely generated but with the property that \(I_{\widehat{\mathbb Z}}G\) is finitely generated over \(\widehat{\mathbb Z}G\). For any integer \(t\), one can consider the probability that \(t\) random elements of \(I_{\widehat{\mathbb Z}}G\) generate \(I_{\widehat{\mathbb Z}}G\) as a \(\widehat{\mathbb Z}G\)-module. In the paper a formula is given, expressing this probability as a function of the structure of the finite simple \(G\)-modules. The author studies profinite groups for which \(I_{\widehat{\mathbb Z}}G\) is \(t\)-generated with positive probability for some positive integer \(t\), calling these profinite groups APFG (Augmentation Positively Finitely Generated); the resonance with PFG profinite groups in name and meaning is evident. It is proved that a countably based profinite group \(G\) is APFG if and only if the number of simple \(G\)-modules of finite order is polynomially bounded as a function of the order. As a consequence, it turns out that PFG groups are APFG; however the author gives examples showing that this is a strict inclusion.
scientific article; zbMATH DE number 6108409