Bernhard Amberg

Groups with a supersoluble triple factorization

In the investigation of factorized groups very often one has to study groups with a triple factorization G=AB=AK=BK, where A and B are subgroups and K is a normal subgroup of G (see, for instance, [2,4, 7, 15, 221). In [3] it was shown that under certain liniteness conditions the triple factorized group G satisfies some nilpotency requirement if A, B, and K satisfy the same nilpotency requirement. In the following, similar statements are proved for some supersolubility conditions.

Subgroups of the adjoint group of a radical ring

It is shown that the adjoint group R of an arbitrary radical ring R has a series with abelian factors and that its finite subgroups are nilpotent. Moreover, some criteria for subgroups of R to be locally nilpotent are given.

Products of groups and group classes

Letχ be a Schunck class, and let the finite groupG=AB=BC=AC be the product of two nilpotent subgroupsA andB andχ-subgroupC. If for every common prime divisorp of the orders ofA andB the cyclic group of orderp is anχ-group, thenG is anχ-group. This generalizes earlier results of O. Kegel and F. Peterson. Some related results for groups of the formG=AB=AK=BK, whereK is a nilpotent normal subgroup ofG andA andB areχ-groups for some saturated formationχ, are also proved.

On finite products of soluble groups

Let the finite groupG =AB be the product of two soluble subgroupsA andB, and letπ be a set of primes. We investigate under which conditions for the maximal normalπ-subgroups ofA, B andG the following holds:Oπ(G) ∩Oπ(G) ⊆Oπ(G).

Products of locally finite groups with min- p

The paper is devoted to showing that if the factorized group G = AB is almost solvable, if A and B are w-subgroups with min-/? for some prime p in w, and also if the hypercenter factor group A/H(A) or B/H(B) has min-/) for every prime />, then G is a w-group with min-p for the prime p.

ON NORMAL SUBGROUPS OF PRODUCTS OF NILPOTENT GROUPS

Let G be a group factorized by finitely many pairwise permutable nilpotent subgroups. The aim of this paper is to find conditions under which at least one of the factors is contained in a proper normal subgroup of G.

Large subgroups of a finite group of even order

It is shown that if G is a group of even order with trivial center such that |G| > 2|CG(t)| for some involution t ∈ G, then there exists a proper subgroup H of G such that |G| < |H|2. If |G| > |CG(t)| and k(G) is the class number of G, then |G| ≤ k(G)3.

Rank formulae for factorized groups

AbstractThe following inequalities for the torsion-free rank r0(G) of the group G=AB and for the p∞-rank rp(G) of the soluble-by-finite group G=AB are stated:

On the Soluble Graph of a Finite Simple Group

\begin{gathered} r_0 (G) \leqslant r_0 (A) + r_0 (B) - r_0 (A \cap B), \hfill \\ r_p (G) \leqslant r_p (A) + r_p (B) - r_p (A \cap B). \hfill \\ \end{gathered}