This contains a new version of the so-called non-commutative Gauss algorithm for polycyclic groups. Its results allow to read off the order and the index of a subgroup in an (possibly infinite) polycyclic group.
We study the class of finite groups $G$ satisfying $Phi (G/N)= Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.
Let $G$ be a simple algebraic group over an algebraically closed field $k$, where $mathrm{char}, k$ is either 0 or a good prime for $G$. We consider the modality $mathrm{mod}(B : mathfrak u)$ of the action of a Borel subgroup $B$ of $G$ on the Lie algebra $mathfrak u$ of the unipotent radical of $B$, and report on computer calculations used to show that $mathrm{mod}(B:mathfrak u) = 20$, when $G$ is of type $mathrm E_8$. This completes the determination of the values for $mathrm{mod}(B:mathfrak u)$ for $G$ of exceptional type.
We investigate the subgroup structure of the hyperoctahedral group in six dimensions. In particular, we study the subgroups isomorphic to the icosahedral group. We classify the orthogonal crystallographic representations of the icosahedral group and analyse their intersections and subgroups, using results from graph theory and their spectra.
The goal of this paper is to give a group-theoretic proof of the congruence subgroup property for $Aut(F_2)$, the group of automorphisms of a free group on two generators. This result was first proved by Asada using techniques from anabelian geometry, and our proof is, to a large extent, a translation of Asadas proof into group-theoretic language. This translation enables us to simplify many parts of Asadas original argument and prove a quantitative version of the congruence subgroup property for $Aut(F_2)$.
Answering a question of Dan Haran and generalizing some results of Aschbacher-Guralnick and Suzuki, we prove that given a set of primes pi, any finite group can be generated by a pi-subgroup and a pi-subgroup. This gives a free product description of a countably generated free profinite group.