6. Normal subgroups with abelian sylow subgroups

Every finite abelian group is a subgroup of the additive group of a finite simple left brace

We extend the definition of a finite -group to profinite groups and give a description of profinite -groups as a triple semidirect product of two prosoluble groups with a semisimple group, extending an old result of A. M. Broshi to the profinite case. We also prove that a profinite -group with finitely generated non-trivial Fitting subgroup is metabelian-by-(finite exponent). If, in addition, G is finitely generated then it is virtually metabelian polycyclic. Communicated by Mandi Schaeffer Fry

An answer to a question of J.G. Thompson on some generalized characters

An automorphism $\\alpha$ of an integral group ring $\\mathbb{Z} G$, where $G$ is a finite group, is said to have a Zassenhaus factorization if it is the composition of an automorphism of $G$ (extended to a ring automorphism) and a central automorphism. In 1988, Roggenkamp and Scott constructed a group $G$ (of order $2880$) such that $\\mathbb{Z} G$ has a normalized (i.e., augmentation preserving) automorphism $\\alpha$ which has no Zassenhaus factorization. In this paper, short proofs of the following two results are given. (1) For a group $G$ of order $144$, there is a normalized automorphism $\\alpha$ of $\\mathbb{Z} G$ which has no Zassenhaus factorization. Moreover, $\\alpha$ can be chosen to have finite order. (2) There is a group $G$ of order $1200$, with abelian Sylow subgroups and Sylow tower, such that $\\mathbb{Z} G$ has a normalized automorphism which has no Zassenhaus factorization.

Let A be a finite nilpotent group acting fixed point freely by automorphisms on the finite solvable group G. It is conjectured that the Fitting length of G is bounded by the number of primes dividing the order of A, counted with multiplicities. The main result of this paper shows that the conjecture is true in the case where A is cyclic of order pnq, for prime numbers p and q coprime to 6 and G has abelian Sylow 2-subgroups.

Let G be a finite group with an abelian normal subgroup N. When does N have a unique conjugacy class of complements in G? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups of G closed under conjugation whose intersection equals Φ ⁢ ( G ) {\Phi(G)} . In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when Φ ⁢ ( G ) = 1 {\Phi(G)=1} , these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.

1. In studying the structure of abstract groups, one is often confronted with the seemingly trivial question: when do we consider a group (or class of groups) as known? The question has many aspects, and is usually a very complex one. Of course, the group is formally known as soon as its multiplication table is determined; but in most cases, little further information can be obtained from the multiplication table itself on the structural elements of the group, such as subgroups, normal subgroups, automorphisms, etc. If one is however, interested in the classification of groups belonging to a certain class, then the problem is concentrated on the enumeration of groups having some common property, and the structural elements of the individual groups are of a secondary importance. In this case, the problem can be formulated in the following way: Find some finite algebraical procedure, depending on some integral or more general algebraical parameters, which enables us to construct, in an abstract or symbolic way, each group belonging to the class under consideration. A group is of course constructed if its elements are known, and the groupproduct of any two elements can be calculated by means of the defining procedure. It is desirable that the parameters, called invariants of the group, should be characteristic of the group in the sense that no two distinct systems of operations should define the same group. A classical example for such an interpretation of the structure problem is furnished by the fundamental theorem of finite Abelian groups. In this paper we are concerned with the enumeration of certain finite metabelian groups. (Metabelian is a group with Abelian commutator subgroup.) The problem in all its generality is a very difficult one, and far from being solved in the above sense. However, special classes of metabelian groups are accessible to investigation, and such investigations have an interest in so far as they clear more aspects of the general problem. In studying the structure of metabelian groups, the method of group extensions is often found to be the most natural tool. As an extensive use will be made of this method in this paper, I want to formulate it in such a way as it seems most suitable for the case discussed here. A similar but formally different formulation was given by A. Scholz1. See also W. Magnus2. Let ( be a finite metabelian group, additively written, 2t an Abelian normal subgroup of (M such that the quotient group e = (5/Wt is Abelian. Elements

The description of the automorphisms of a unipotent subgroup U of a Chevalley group over a field K known earlier for charK ≠ 2, 3 (Gibbs, 1970) was completed in 1990 together with the solution of problem (1.5) from A.S. Kondrat’ev’s survey (Usp. Mat. Nauk, 1986). In the present paper, Aut U is described for the case of finitary Chevalley groups. For a Chevalley group of classical type over a finite field, it is proved that any large Abelian subgroup from U is conjugate to a normal subgroup in U. It is shown that this is not so in the general case; therefore, problem (1.6) from Kondrat’ev’s survey about large Abelian subgroups in U is reduced to listing the exceptions. The orders of these subgroups and large Abelian normal subgroups in U were listed earlier.

It is shown that if a group G = A B G = AB , where A A and B B are Abelian subgroups of G , A ≠ B G,A \ne B , and either A A or B B satisfies the maximum condition, then there is a normal subgroup N N of G G , N ≠ G N \ne G , such that N N contains either A A or B B .

It is shown that if a group G=AB, where A and B are Abelian subgroups of G, A^B, and either A or B satisfies the maximum condition, then there is a normal subgroup N of G, N^G, such that N contains either A or B. Summary and background.The purpose of this paper is to show that, if a group G is a nontrivial product of two Abelian subgroups A and B, and if either A or B satisfies the maximum condition, then there is a proper normal subgroup of G containing either A or B. (L.E. Knop [7]).The basic impetus for this research is found in a paper of N. Itô [1].In this paper, Itô showed that if G is a product of two Abelian subgroups A and B, then G is metabelian.Furthermore, he showed that if G is finite, then there is a normal subgroup of G, not the identity, which is contained in either A or B, and also that there is a proper normal subgroup of G which contains either A or B.Additional work on the problem of when there is a normal subgroup contained in one of the two factors has been done by P. M. Cohn [2], who extended Itô's result to the case where A and B are infinite cyclic, and N. F. Sesekin [3], who extended the result to the case where A and B are finitely connected.E. Schenkman [4] claimed that both of Itô's results could be extended to the case where both A and B satisfied the maximum condition.Unfortunately there was a flaw in the proof that either A or B contained a normal subgroup of G (discovered by both Schenkman, Zentralblatt review [4], and W. R. Scott [5]), and the proof that either A or B was contained in a proper normal subgroup of G was based on this result.B. Amberg and W. R. Scott [6, Theorem 4] did show that if either A or B

In [6] B. H. Neumann proved the following beautiful result:if a group G is covered by finitely many cosets, say G=xiHi, then we can omit from the union any xiHi, for which|G|Hj|is infinite. In particular, |G:Hj|is finite, for some j∈ {l,…,n}.In an unpublished result R. Baer characterized the groups covered by finitely many abelian subgroups, they are exactly the centre-by-finite groups [8]. Coverings by nilpotent subgroups or by Engel subgroups and by normal subgroups have been studied, for example, by R. Baer (see [8]), L. C. Kappe [2,1], M. A. Brodie and R. F. Chamberlain [1], and recently by M. J. Tomkinson [9].

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.

Injective modules, localization and completion in group algebras

Let A A be an abelian subgroup of maximal order in the finite metabelian p p -group P P . It is shown that there exists a normal abelian subgroup A 1 {A_1} of P P such that the order of A 1 {A_1} is equal to the order of A A .

On a class of inseparable finite groups