2020/09/19 by Snopce, Ilir, Tanushevski, Slobodan
#12F10 (Primary) 12G05 #20E18 #22E20 (Secondary) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2009.09297
We call a pro-p group G Frattini-injective if distinct finitely generated subgroups of G have distinct Frattinis. This paper is an initial effort toward a systematic study of Frattini-injective pro-p groups (and several other related concepts). Most notably, we classify the p-adic analytic and the solvable Frattini-injective pro-p groups, and we describe the lattice of normal abelian subgroups of a Frattini-injective pro-p group. We prove that every maximal pro-p Galois group of a field that contains a primitive pth root of unity (and also contains √(-1) if p=2) is Frattini-injective. In addition, we show that many substantial results on maximal pro-p Galois groups are in fact consequences of Frattini-injectivity. For instance, a p-adic analytic or solvable pro-p group is Frattini-injective if and only if it can be realized as a maximal pro-p Galois group of a field that contains a primitive pth root of unity (and also contains √(-1) if p=2); and every Frattini-injective pro-p group contains a unique maximal abelian normal subgroup.