Some topics on finite p-groups and pro-p groups

Abstract

This thesis consists of three parts, each of them devoted to different aspects of the theory of finite p-groups and pro-p groups. The first part is concerned with the study of the following problem: under which conditions on a group G does the verbal subgroup for a given word w coincide with the set of w-values? We will analyse this problem for different words lying in the derived subgroup of the free group, namely, the commutator word, the lower central words and general outer commutator words, under the hypothesis that G is a finite p-group.The second part is aimed to the study of the Hausdorff dimension function. In recent decades, this fractal dimension has provided interesting and fruitful applications in the context of profinite groups, all of them based on the pioneering formula by Barnea and Shalev, according to which the Hausdorff dimension of a closed subgroup H of a profinite group G can be regarded as the “logarithmic density” of H in G. Thus, we will focus on the notion of normal Hausdorff spectrum of G with respect to a given filtration series, giving the first example of a finitely generated pro-p group with full normal Hausdorff spectra.Finally, in the third part of the thesis, we will introduce two new classes of power-ful p-groups: the powerfully solvable groups and the powerfully simple groups. These are powerful p-groups that somehow fulfil the “role” that finite solvable groups and finite simple groups have in the class of all finite groups, respectively. We will provide some results and classification concerning these groups, including a Jordan-Hölder type theorem. For this purpose, a bijective correspondence between the category of certain powerful groups and the category of alternating algebras over Fp will be of particular interest.