Class Of Soluble Groups Research Articles

Sylow like theorems for $V(mathbb{Z}G)$

‎The main part of this article is a survey on torsion subgroups of the unit group of‎ ‎an integral group ring‎. ‎It contains the major parts of my talk given at the‎ ‎conference‎ ‎Groups,‎ ‎Group Rings and Related Topics‎ ‎at UAEU in Al Ain October 2013‎. ‎In the second part special emphasis is layed on $p$‎ - ‎subgroups and on the‎ ‎open question whether there is a Sylow like theorem in the‎ ‎normalized unit group of an integral group ring‎. ‎For specific classes of finite groups we prove that $p$‎ - ‎subgroups‎ ‎of the normalized unit group of its integral group rings $V(mathbb{Z}G)$ ‎are isomorphic to subgroups of $G‎ .‎$ In particular for $p = 2$ this is shown ‎when $G$ has abelian Sylow $2$‎ - ‎subgroups‎. ‎This extends results known‎ ‎for soluble groups to classes of groups which are not contained in the‎ ‎class of soluble groups‎.

Free subgroups in division rings generated by group rings of soluble groups

We study free subgroups in the multiplicative group of division rings generated by group rings of some classes of soluble groups.

Groups with poly-context-free word problem

We consider the class of groups whose word problem is poly-context-free ; that is, an intersection of finitely many context-free languages. We show that any group which is virtually a finitely generated subgroup of a direct product of free groups has poly-context-free word problem, and conjecture that the converse also holds. We prove our conjecture for several classes of soluble groups, including metabelian groups and torsion-free soluble groups, and present progress towards resolving the conjecture for soluble groups in general. Some of the techniques introduced for proving languages not to be poly-context-free may be of independent interest.

ON A PERMUTABILITY PROPERTY OF SUBGROUPS OF FINITE SOLUBLE GROUPS

The structure and embedding of subgroups permuting with the system normalizers of a finite soluble group are studied in the paper. It is also proved that the class of all finite soluble groups in which every subnormal subgroup permutes with the Sylow subgroups is properly contained in the class of all soluble groups whose subnormal subgroups permute with the system normalizers while this latter is properly contained in the class of all supersoluble groups.

Pairwise -connected Products of Certain Classes of Finite Groups#

Subgroups A and B of a finite group are said to be 𝒩-connected if the subgroup generated by elements x and y is a nilpotent group, for every pair of elements x in A and y in B. The behaviour of finite pairwise permutable and 𝒩-connected products are studied with respect to certain classes of groups including those groups where all the subnormal subgroups permute with all the maximal subgroups, the so-called SM-groups, and also the class of soluble groups where all the subnormal subgroups permute with all the Carter subgroups, the so-called C-groups.

An Amalgamation Theorem For Soluble Groups

AbstractA theorem of G. Higman about the embeddability of amalgams within the class of all finite ρ-groups is generalized to classes of soluble groups. We also give best possible bounds for the solubility lengths of the constructed completions. And, as an application, the super-soluble amalgamation bases in the class of all finite soluble π-groups are determined.

A class of soluble groups

In this paper a problem in group theory is solved to produce an interesting class of groups. The motivation, however, comes from algebraic number theory and in particular the Artin representation. To describe this representation we follow [5, Chap. IV and VI]. Let K and L be local fields, i.e., fields complete with respect to a discrete valuation and with perfect residue field. Let L have valuation ring A, and valuation v, extending the valuation on K. Suppose that L/K is a finite Galois extension with Galois group H. Then H has a normal series consisting of the ramification groups Hj = {O E H: V~(UX X) > j + 1 V x E A,}, j = -1, 0, I,2 ,..., which produces a character of H, due to Artin, as described in Theorem 1 below. The notation adopted in this paper is as follows. If G is a finite group, then Irr(G) denotes its set of irreducible characters. l,, rG, and U, = rc 1, are its principal, regular, and augmentation characters, respectively, whilst unless otherwise stated xc and xG will mean the restriction of x to G and the character of G induced from x.

Conjugacy separability and separable orbits

Two elements (subgroups) of a group G are called conjugacy separable if they are conjugate in G if and only if their images are conjugate in every finite quotient of G. The whole group is termed conjugacy separable if each pair of its elerr,ents is conjugacy separable. Conjugacy separable groups form a rather complicated class of groups. It is closed with respect to forming free products but not under taking subgroups, forming extensions and wreath products [S]. Restriction to the class of soluble groups does not help either. The best known result, the Theorem of Formanek [l] and Remeslennikov [7] yields that this class contains all polycyclic-by-finite groups. These groups also have all subgroups conjugacy separable (Grunewald, Segal [2]). In this paper we will prove conjugacy separability of elements and subgroups for a class of not necessarily finitely generated nilpotent groups of finite abelian section rank. Some examples will show that there is nearly no general way to extend these results to a wider class of groups. Regarding conjugacy as an operation of a group on itself yields another way how to extend theorems on conjugacy separability, which I found in a recent paper of Hilton and Roitberg [3]. Let Q be a group operating on a further group G. Two elements (subgroups) of G are said to bl: Q-conjugate, if there exists an element qE Q mapping the first element (subgr/Jup) onto the second. They are termed Q-separable if they are Q-conjugate or if there e!xists a finite Q-quotient of G in which their images are net Q-conjugate. G has separable Q-orbits if each pair of its elements is Q-separable. As usual Q is said to act nh’potently on G if it acts identically on the factors of a finite Q-invariant series of G and almost nilpotently if it conta:lns a subgroup of finite index acting nilpoltently on a Q-invariant subgroup of finite index of G. Hilton and Roitberg proved orbit separability for finitely generated nilpotent groups on which a finitely generated nilpotent group acts nilpotently. We will extend

Subgroup closed Fitting classes are formations

Since their introduction by Fischer(12) and Fischer, Gaschütz and Hartley (13) Fitting classes of soluble groups have attracted attention on two fronts (all groups considered in this paper will be finite and soluble). On the one hand is their important role in the structure of finite soluble groups, a good account of which can be found in Gaschütz (14), and on the other is their intrinsic interest as classes of groups. This paper falls into the second category, and is a continuation and completion of (8). There we proved that a subgroup closed Fitting class is a formation if it consists of groups of nilpotent length at most three. Happily, at last, we can remove this qualification.

An example in the theory of solubl groups

Let G be a finite soluble group. In (1) Alperin proves that two system normalizers of G contained in the same Carter subgroup C of G are conjugate in C. In recent unpublished work G.A.Chambers of the University of Wisconsin has proved that, if is a saturated formation, the -normalizers of an A-group are pronormal subgruops; hence, in particular, that two -normalizers contained in an -projector E of an A-group are conjugate in E. In this note we describe an example which shows that in Alperin's theorem the class of nilpotent groups cannot in general be replaced by an arbitary saturated formation without some restriction on the class of soluble groups under consideration. we provePROPOSITION. There exists a saturated formationand a group G which has two-normalizers E1and E2contained in an-projector F of G such that E1and E2are not conjugate in F.

A note on system normalizers of a finite soluble group

Introduction. Hall ((3), (4)) introduced the concept of a Sylow system and its normalizer into the theory of finite soluble groups. In (4) he showed that system normalizers may be characterized as those subgroups D of G minimal subject to the existence of a chain of subgroups from D up to G in which each subgroup is maximal and non-normal in the next; he also showed that a system normalizer covers all the central chief factors and avoids all the eccentric chief factors of G (for definitions of covering and avoidance, and an account of their elementary properties), the reader is referred to Taunt ((5)). This note arises out of an investigation into the question to what extent this covering/ avoidance property characterizes system normalizers; it provides a partial answer by means of two elementary counter-examples given in section 3 which seem to indicate that the property ceases to characterize system normalizers as soon as the ‘non-commutativity’ of the group is increased beyond a certain threshold. For the sake of completeness we include proofs in Theorems 1 and 2 of generalizations of two known results communicated to me by Dr Taunt and which as far as we know have not been published elsewhere. Theorem 1 shows the covering/avoidance property to be characteristic for the class of soluble groups with self-normalizing system normalizers introduced by Carter in (1), while Theorem 2 shows the same is true for A -groups (soluble groups with Abelian Sylow subgroups investigated by Taunt in (5)).