Omega subgroups of pro-p groups

On p-saturable groups

“…Now, the lemma follows from the fact that finite p-groups are nilpotent. The second part of the lemma is a particular case of Theorem 2.5 of [2]. 2…”

“…More recently (see [2]), a new family of pro-p groups has been defined: a closed normal subgroup N of a pro-p group G is a PF-embedded in G if there exists a central series of subgroups {N i } i∈N starting at N with trivial intersection, and with the property that [N i , G, p−1 . .…”

“…This new family of subgroups has been successfully used to study the power structure of pro-p groups. For example, it has been proved in [2] that the torsion elements of a finitely generated pro-p group with the condition γ h(p−1) (G) G p h+1 form a finite subgroup (this result was already proven for p > 2 by Wilson in [10], but the bounds on the exponent of the torsion subgroup were not precise enough). One of the most remarkable facts about PF-embedded subgroups is that have they a very nice power-commutator structure inside the group.…”

On $p$-class groups and the Fontaine-Mazur conjecture

“…is called geometric, if it is unramified outside a finite set of primes of k and its restriction ρ |G p to the decomposition group G p of p is potentially semi-stable for all primes p of k dividing p. 3 In [5], Fontaine and Mazur make the following fundamental conjecture: ρ is geometric if and only if it comes from algebraic geometry, i.e., it arises from the Galois action on anétale cohomology group H i et (V k , Q p (j)) of a smooth projective variety V over k. It has been proven in the GL 2 -case (under some further assumptions) by Kisin (cf. [9]).…”

Finite p-central groups of height k

“…Let p e be the exponent of Ω 1 (G, p). By the Hall-Petrescu collection formula (see [8,Theorem 2.1]), for any y ∈ Ω 1 (G, p) and x ∈ G one has (3.10) [y,…”