Two properties of the filtration of the outer automorphism groups of certain groups
Let \(G\) be a finitely generated discrete or pro-\(\ell\)-group, \(\ell\) being a fixed prime number. Let \(\Gamma\) denote the outer automorphism group of \(G\). The group \(\Gamma\) has a filtration \(\{\Gamma[m]\}_{m \geq 0}\) which is naturally induced from the descending central series of \(G\). We shall show, under some assumptions on \(G\), that the abelian group \(\Gamma[m]/\Gamma[m+1]\) (\(m \geq 1\)) is a free \(Z\) (or \(Z_ \ell\))- module of finite rank and that the intersection \(\bigcap^ \infty_{m = 0} \Gamma[m]\) reduces to \(\{1\}\) (in the pro-\(\ell\) case).
scientific article; zbMATH DE number 737164
scientific article