Free-by-finite pro-\(p\) groups and integral \(p\)-adic representations
The paper under review is a contribution to the theory of integral \(p\)-adic representations of finite \(p\)-groups. Given a free pro-\(p\) group \(F_n\) of finite rank \(n\), the natural epimorphism of \(F_n\) onto its quotient over the closure of its derived subgroup induces an epimorphism of the group \(\Aut(F_n)\) of automorphisms of \(F_n\) onto \(\text{GL}_n(\mathbb Z_p)\), where \(\mathbb Z_p\) is the ring of \(p\)-adic integers. In the main results, the authors find out which integral \(p\)-adic representations \(\rho\colon H\to\text{GL}_n(\mathbb Z_p)\) lift to a representation \(H\to\Aut(F_n)\), in the cases when \(H\) is a finite cyclic \(p\)-group, or \(H\) is a finite \(p\)-group such that for every subgroup \(C\) of \(H\) the restriction of \(\rho\) to \(C\) has no trivial indecomposable components.
\(p\)-adic representations of finite groups
scientific article; zbMATH DE number 5963415