On the structure of normal subgroups of potent p-groups

Zeta function of representations of compact 𝑝-adic analytic groups

“…More results in this direction can be found in [17]. It follows implicitly from [10] that any normal subgroup of a uniform pro-p group is strong if p > 2. In Lemma 2.1 we present a family of strong pro-p groups which appear naturally in this work.…”

Omega subgroups of pro-p groups

“…Thus G is trivial for p = 2 and G is potent if p > 2. In the latter case, the result that Ω i (G) p i = 1 has been established in Theorem 1.1 of [7].…”

“…In some circumstances, these sets of generators already form a subgroup, in other words, every element of G p i is a p i -th power and the exponent of Ω i (G) is at most p i . This is the case of regular finite p-groups, as P. Hall showed in his pioneering work [9], and of potent pro-p groups for odd p, as proved by Arganbright [1] for power subgroups and by González-Sánchez and Jaikin-Zapirain [7] for omega subgroups. Recall that for p > 2 a pro-p group is potent if γ p−1 (G) ≤ G p .…”

“…It is inspired by a similar result for potent groups, see Theorem 4.1 in [7]. Theorem 3.6: Let G be a pro-p group.…”

On p-saturable groups

“…Since G is a potent group, γ p (G) [G p , G] = [G, G] p (see Theorems 3.1 and 3.2 of [3]). Therefore, by Theorem 3.4, G is p-saturable and applying Theorem 6.1 of [3] we have that all normal subgroups of G satisfy the condition of the previous corollary. 2…”