An analogue of the Nielsen-Schreier formula for pro-p-groups

Abstract

We introduce a condition analogous to the Nielsen-Schreier formula, and investigate basic properties of pro-p-groups satisfying the condition.

Pro-p groups with constant generating number on open subgroups

“…Note that E n is non-empty for every n because, clearly, it contains the free abelian pro-p group Z n p of rank n. In [11], Yamagishi remarked that no other examples are known to him when n ≥ 3. Recently, for p > 3, the second author determined all p-adic analytic pro-p groups that belong to the class E 3 ; see [10].…”

Pro-p groups of rank 3 and the question of Iwasawa

“…Note that E n is nonempty for n ≥ 1, because there is a trivial example G = Z n p which obviously belongs to E n . In [8] Yamagishi remarked that no other examples are known when n ≥ 3.…”

Corrigendum The number of generators of finite p-groups

“…Finally, we mention that the paper [5] considers pro-p groups G satisfying the inequality dðHÞ À n c jK : HjðdðKÞ À nÞ, where H and K are open subgroups of G and n is some positive integer. For n > 1, such groups lie in M p .…”