A classical and very useful result in the theory of finite groups says that every proper subgroup H of a finite p-group G is strictly contained in its normalizer. Since the appearance of finitely generated, infinite, simple p-groups, such as Olshanskii's Tarski monsters, it became clear that this result cannot be generalized to all finitely generated p-groups. However, it is an intriguing open problem whether the finiteness assumption on G can be replaced by the much weaker assumption that the p-group G is finitely generated and residually finite. A slightly different formulation of this problem was provided by Passman, who asked whether every maximal subgroup of a finitely generated, residually finite p-group is normal in G.

The aim of this project is to give a negative answer to Passman's question. In order to do so, the strategy is to construct a finitely generated, residually finite p-group H that admits a finitely generated proper subgroup G such that (G,H) is a Grothendieck pair, i.e. the inclusion map from G to H extends to an isomorphism on the profinite completion of G and H. It is not difficult to see that such a pair would indeed provide a negative example to Passman's question.

In fact, it is possible to extract Grothendieck pairs (G,H) from a recent work of Steffen Kionke and myself where G and H are finitely generated torsion groups.

However, it seems that the latter groups G and H cannot be chosen to be p-groups by only adjusting the techniques from our work. The goal of my stay is to overcome this issue by choosing G and H to be appropriate branch groups. In particular I am interested in the case where G is the Grigorchuk group. One advantage of this case is that the profinite completion of G was already determined by Goulnara Arzhantseva and Zoran Sunic.