Infinite generation of automorphism groups of free pro-\(p\) groups
The author considers the automorphism group of a free pro-\(p\) group as well as the automorphism group of a free profinite group \(F_ n\). The results concern uniformly saturated pro-\(p\) groups. It is proved that if \(G_ 0\) is a saturated pro-\(p\) group, then the completed group algebra \(\widehat {S} =Z/_ p ZG_ 0\) satisfies the maximal condition for closed right ideals. The main theorem states that the automorphism group of the free metabelian pro-\(p\) group \(M_ n (p)\) of rank \(n \geq 2\) is infinitely generated (i.e. there is no finite generating set, in the topological sense). From that it follows as a corollary, that the automorphism group of the free pro-\(p\) group \(F_ n (p)\) of rank \(n \geq 2\) is infinitely generated and the same for the automorphism group \(\text{Aut }F_ n\) of the free profinite group \(F_ n\) of finite rank \(n \geq 2\). Another application of the results of the author is that if \(N_{n,m}\) (resp. \(M_{n,m}\)) is the free nilpotent group (free metabelian nilpotent group) of rank \(n\) and class \(m\), then the number of generators for the groups \(\text{Aut }N_{n,m}\) and \(\text{Aut }M_{n,m}\) increases without bound as \(m\) increases for \(n \geq 2\). The proofs are described via \(p\)-adic analytic groups. Details of the terminology and symbolism are difficult to be given here.
scientific article; zbMATH DE number 652183