A subset X of a finite group G is said to be prime-power-independent if each element in has prime power order and there no proper Y with 〈Y,Φ(G)〉=〈X,Φ(G)〉, where Φ(G) the Frattini subgroup G. Bpp all generating sets for have same cardinality. We prove that, Bpp, then solvable. Pivoting on some recent results Krempa Stocka (2014); (2020), this yields complete classification Bpp-groups.

let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.

We define the toplogical generating rank dtop(G) of a topological group G as the minimal number of elements of G which generate a dense subgroup of G. I will present joint work with Gena Noskov, partly in progress, about our attempts to compute the topological generating rank for connected Lie groups. The cases that G is semisimple, where dtop(G) is slightly bigger than one, and that G is nilpo...

Algorithms are constructed which, when an explicit presentation of a finitely generated metabelian group G in the variety si1 is given, produce unitary presentations for the derived subgroup G', the centre Z{G), the Fitting subgroup Fit(G), and the Frattini subgroup <p(G). Additional algorithms of independent interest are developed for commutative algebra which construct the associated set of p...

Recently the first example of a family pro-p groups, for p prime, with full normal Hausdorff spectrum was constructed. In this paper we further investigate by computing their finitely generated respect to each five standard filtration series: p-power series, iterated lower p-series, Frattini series and dimension subgroup series. Here spectra these groups consist infinitely many p-adic rational ...