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1.
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). 相似文献
2.
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 相似文献
3.
Jean Ludwig 《Monatshefte für Mathematik》1986,101(2):159-165
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. 相似文献
4.
《代数通讯》2013,41(5):1895-1902
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. 相似文献
5.
6.
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 GLn(D) satisfying a generalized group identity over GLn(D) is central. 相似文献
7.
Charles Lanski 《代数通讯》2013,41(5):1427-1446
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. 相似文献
8.
Gemma Parmeggiani 《Rendiconti del Circolo Matematico di Palermo》1995,44(2):215-238
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. 相似文献
9.
《代数通讯》2013,41(9):3367-3373
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. 相似文献
10.
Ryutaroh Matsumoto 《代数通讯》2013,41(1):401-405
In Hai and Thin [1], there is a theorem, stating that every locally nilpotent subnormal subgroup in a division ring D is central (see [1, 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. 相似文献
11.
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 相似文献
12.
Bijan Taeri 《代数通讯》2013,41(3):894-922
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. 相似文献
13.
Jung Wook Lim 《代数通讯》2015,43(1):345-356
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. 相似文献
14.
Gerhard RÖhrle 《Geometriae Dedicata》1997,66(1):51-64
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. 相似文献
15.
Gyu Whan Chang 《代数通讯》2013,41(9):3309-3320
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. 相似文献
16.
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. 相似文献
17.
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. 相似文献
18.
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)] < ∞. 相似文献
19.
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. 相似文献
20.
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. 相似文献