The number of generators of ideals in graded group algebras and \(p\)-adic analytic pro-\(p\) groups of dimension one (Q1340178)

Let \(G\) be any group, and let \(K\) be a field of characteristic \(p > 0\). It is shown that the number of generators of ideals in the graded group ring \(\text{gr}(KG)\) is finitely bounded precisely when the pro-\(p\) completion of \(G\) is \(p\)-adic analytic of dimension at most one. Graded restricted universal enveloping algebras are examined in a similar fashion.