On automorphisms of free pro-𝑝-groups. I

Abstract

Let F F be a (topologically) finitely generated free pro- p p -group, and β \beta an automorphism of F F . If p ≠ 2 p \ne 2 and the order of β \beta is 2, then there is some basis of F F such that β \beta either fixes or inverts its elements. If p p does not divide the order of β \beta , then the subgroup of F F of all elements fixed by β \beta is (topologically) infinitely generated; however this is not always the case if p p divides the order of β \beta .