Maximal Subgroups of Subnormal Subgroups of GL n (D) with Finite Conjugacy Classes

Let D be an infinite division ring. A famous result due to Herstein says that every noncentral element of D has infinitely many conjugates in D. So, if D* is an FC-group, then D is a field. Now let N be a subnormal subgroup of GL _n (D), where n ≥ 1, and M a maximal subgroup of N. In this article, we prove that if M is an FC-group, then M is contained in the multiplicative group of some subfield of M _n (D).     

A Criterion for the Triviality of G(D) and Its Applications to the Multiplicative Structure of D

Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/Nrd _D/F (D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK ₁(D) = 1 and F ^*2 = F ^*2n . Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated     

Prime ideals in the C*-algebra of a nilpotent group

We say that a locally compact groupG hasT ₁ primitive ideal space if the groupC ^*-algebra,C ^*(G), has the property that every primitive ideal (i.e. kernel of an irreducible representation) is closed in the hull-kernel topology on the space of primitive ideals ofC ^*(G), denoted by PrimG. This means of course that every primitive ideal inC ^*(G) is maximal. Long agoDixmier proved that every connected nilpotent Lie group hasT ₁ primitive ideal space. More recentlyPoguntke showed that discrete nilpotent groups haveT ₁ primitive ideal space and a few month agoCarey andMoran proved the same property for second countable locally compact groups having a compactly generated open normal subgroup. In this note we combine the methods used in [3] with some ideas in [9] and show that for nilpotent locally compact groupsG, having a compactly generated open normal subgroup, closed prime ideals inC ^*(G) are always maximal which implies of course that PrimG isT ₁.     

Subgroups of GL 1(R) for Local Rings R

Abstract

Let R be a local ring, with maximal ideal m , and residue class division ring R/ m ?=?D. Denote by R*?=?G L ₁(R), the group of units of R. Here we investigate some algebraic structure of subnormal and maximal subgroups of R*. For instance, when D is of finite dimension over its center, it is shown that finitely generated subnormal subgroups of R* are central. It is also proved that maximal subgroups of R* are not finitely generated. Furthermore, assume that P is a nonabelian maximal subgroup of R* such that P contains a noncentral soluble normal subgroup of finite index, it is shown that D is a crossed product division algebra.     

Some Skew Linear Groups Satisfying Generalized Group Identities

In this article, we consider a type of generalized group identity and extend some earlier results. For example, we show that, if D is a division ring with infinite center, then every subnormal subgroup of GL_n(D) satisfying a generalized group identity over GL_n(D) is central.     

Derivations nilpotent on subsets of prime rings

ABSTRACT

Let D be a division algebra with center F. Consider the group CK ₁(D) = D*/F*D′ where D* is the group of invertible elements of D and D′ is its commutator subgroup. In this note we shall show that, assuming a division algebra D is a product of cyclic algebras, the group CK ₁(D) is trivial if and only if D is an ordinary quaternion algebra over a real Pythagorean field F. We also characterize the cyclic central simple algebras with trivial CK ₁ and show that CK ₁ is not trivial for division algebras of index 4. Using valuation theory, the group CK ₁(D) is computed for some valued division algebras.

     

Soluble (HN)2-groups

In this paper we investigate the class of finite soluble groups in which every subnormal subgroup has normal normalizer. In particular we prove that they areUN ₂U, whereU andN ₂denote finite abelian groups and of finite nilpotent groups of class at most 2 respectively.     

Irreducible Subgroups of GL 1(D) Satisfying Group Identities

ABSTRACT

Let D be a finite dimensional F -central division algebra and G an irreducible subgroup of D*: = GL ₁(D). Here we investigate the structure of D under various group identities on G. In particular, it is shown that when [D:F] = p ², p a prime, then D is cyclic if and only if D* contains a nonabelian subgroup satisfying a group identity.     

On computing the integral closure

In Hai and Thin [1 Hai , B. X. , Thin , N. V. On locally nilpotent subgroups of GL ₁(D). Communications in Algebra 37 ( 2 ): 712 – 718 . [Google Scholar]], there is a theorem, stating that every locally nilpotent subnormal subgroup in a division ring D is central (see [1 Hai , B. X. , Thin , N. V. On locally nilpotent subgroups of GL ₁(D). Communications in Algebra 37 ( 2 ): 712 – 718 . [Google Scholar], Theoerem 2.2]). Unfortunately, there is some mistake in the proof of this theorem. In this note, we give the another proof of this theorem.     

A Generalization of <Emphasis Type="Italic">T</Emphasis>-Groups

This paper identifies a certain class of locally supersoluble groups (called soluble hall-T groups) which contains the soluble T-groups as well as the nilpotent groups. The main result states that the product of a normal soluble hall-T subgroup and a subnormal locally supersoluble subgroup is always locally supersoluble.AMS Subject Classification (1991): 20E25, 20F16, 20F19     

Characterization of 3-Rewritable Finite Nilpotent Groups

Let n be an integer greater than 1. A group G is said to be “ n-rewritable” whenever for every n elements x ₁,…,x _n of G, there exist distinct permutations τ, σ on the set {1,2,…, n} such that x _τ(1) ··· x _τ(n) = x _{σ (1)} ··· x _{σ (n)}. In this article, we complete the classification of 3-rewritable finite nilpotent groups and prove that a finite nilpotent group G is 3-rewritable if and only if G has an abelian subgroup of index 2 or the derived subgroup has order < 6.     

A General Theory of Almost Splitting Sets

Let * be a star-operation of finite type on an integral domain D. In this paper, we generalize and study the concept of almost splitting sets. We define a saturated multiplicative subset S of D to be an almost g*-splitting set of D if for each 0 ≠ d ∈ D, there exists an integer n = n(d) ≥1 such that d ⁿ  = st for some s ∈ S and t ∈ D with (t, s′)_* = D for all s′ ∈ S. Among other things, we prove that every saturated multiplicative subset of D is an almost g*-splitting set if and only if D is an almost weakly factorial domain (AWFD) with *-dim (D) = 1. We also give an example of an almost g*-splitting set which is not a g*-splitting set.     

Maximal Parabolic Subgroups in Classical Groups are of Modality Zero

The principal aim of this paper is to show that every maximal parabolic subgroup P of a classical reductive algebraic group G operates with a finite number of orbits on its unipotent radical. This is a consequence of the fact that each parabolic subgroup of a group of type A _n whose unipotent radical is of nilpotent class at most two has this finiteness property.     

Characterizations of *-Cancellation Ideals of an Integral Domain

Let D be an integral domain and * a star-operation on D. For a nonzero ideal I of D, let I ^{* _f} = ?{J* | (0) ≠ J ? I is finitely generated} and I ^{* _w} = ? _{P∈* f -Max(D)} ID _P . A nonzero ideal I of D is called a *-cancellation ideal if (IA)* = (IB)* for nonzero ideals A and B of D implies A* =B*. Let X be an indeterminate over D and N _* = {f ∈ D[X] | (c(f))* =D}. We show that I is a * _w -cancellation ideal if and only if I is * _f -locally principal, if and only if ID[X] _{N *} is a cancellation ideal. As a corollary, we have that each nonzero ideal of D is a * _w -cancellation ideal if and only if D _P is a principal ideal domain for all P ∈ * _f -Max(D), if and only if D[X] _{N *} is an almost Dedekind domain. We also show that if I is a * _w -cancellation ideal of D, then I ^{* _w}  = I ^{* _f}  = I _t , and I is * _w -invertible if and only if I ^{* _w}  = J _v for a nonzero finitely generated ideal J of D.     

A note on the existence of non-cyclic free subgroups in division rings

A division ring D is said to be weakly locally finite if for every finite subset ${S \subset D}$ , the division subring of D generated by S is centrally finite. It is known that the class of weakly locally finite division rings strictly contains the class of locally finite division rings. In this note we prove that every non-central subnormal subgroup of the multiplicative group of a weakly locally finite division ring contains a non-cyclic free subgroup. This generalizes the previous result by Gonçalves for centrally finite division rings.     

On maximal subgroups of the multiplicative group of a division algebra

The question of existence of a maximal subgroup in the multiplicative group D^* of a division algebra D finite-dimensional over its center F is investigated. We prove that if D^* has no maximal subgroup, then deg(D) is not a power of 2, F^*2 is divisible, and for each odd prime p dividing deg(D), there exist noncyclic division algebras of degree p over F.     

Maximal subgroups of <Emphasis Type="Italic">GL</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">D</Emphasis>) with finite conjugacy classes

Let D be an infinite division ring. A famous result due to Herstein says that every non-central element of D has infinitely many conjugates and so, if D ^* is an FC-group, then D is a field. Let M be a maximal subgroup of GL _n (D), where n ≥ 1. In this paper, we prove that if M is an FC-group, then it is the multiplicative group of some maximal subfield of M _n (D). Moreover, if M is algebraic over Z(D), then [D : Z(D)] < ∞.     

On Influence of Contranormal Subgroups on the Structure of Infinite Groups

Following Rose, a subgroup H of a group G is called contranormal, if G = H ^G . In certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. However, for the infinite groups this criterion is not valid. There are examples of non-nilpotent infinite groups whose subgroups are subnormal; in paricular, these groups have no contranormal subgroups. Nevertheless, for some classes of infinite groups, the absence of contranormal subgroups implies the nilpotency of the group. The current article is devoted to the search of such classes. Some new criteria of nilpotency in certain classes of infinite groups have been established.     

Supersoluble crossed product criterion for division algebras

LetD be a finite-dimensionalF-central division algebra. A criterion is given forD to be a supersoluble (nilpotent) crossed product division algebra in terms of subgroups of the multiplicative groupD* ofD. More precisely, it is shown thatD is a supersoluble (nilpotent) crossed product if and only ifD* contains an abelian-by-supersoluble (abelian-by-nilpotent) generating subgroup.