A pro-2 group with full normal Hausdorff spectra

We construct a $2$-generated pro-$2$ group with full normal Hausdorff spectrum $[0,1]$, with respect to each of the four standard filtration series: the $2$-power series, the lower $2$-series, the Frattini series, and the dimension subgroup series. This answers a question of Klopsch and the second author, for the even prime case; the odd prime case was settled by the first author and Klopsch. Also, our construction gives the first example of a finitely generated pro-$2$ group with full Hausdorff spectrum with respect to the lower $2$-series.Comment: 14 page

The finitely generated Hausdorff spectra of a family of pro-pp groups

Recently the first example of a family of pro-$p$ groups, for $p$ a prime, with full normal Hausdorff spectrum was constructed. In this paper we further investigate this family by computing their finitely generated Hausdorff spectrum with respect to each of the five standard filtration series: the $p$-power series, the iterated $p$-power series, the lower $p$-series, the Frattini series and the dimension subgroup series. Here the finitely generated Hausdorff spectra of these groups consist of infinitely many rational numbers, and their computation requires a rather technical approach. This result also gives further evidence to the non-existence of a finitely generated pro-$p$ group with uncountable finitely generated Hausdorff spectrum.Comment: 25 page

Powerfully solvable and powerfully simple groups

We introduce the notion of a powerfully solvable group. These are powerful groups possessing an abelian series of a special kind. These groups include in particular the class of powerfully nilpotent groups. We will also see that for a certain rich class of powerful groups we can naturally introduce the term powerfully simple group and prove a Jordan-H\"older type theorem that justifies the term.Comment: 19 page

Lower central words in finite p-groups

It is well known that the set of values of a lower central word in a group G need not be a subgroup. For a fixed lower central word γr and for p ≥ 5, Guralnick showed that if G is a finite p-group such that the verbal subgroup γr(G) is abelian and 2-generator, then γr(G) consists only of γr-values. In this paper we extend this result, showing that the assumption that γr(G) is abelian can be dropped. Moreover, we show that the result remains true even if p= 3. Finally, we prove that the analogous result for pro-p groups is true