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.     

On sse-embedded subgroups of finite groups

Abstract

In this paper, we introduce the concept of sse-embedded subgroups of finite groups and present some new characterizations of solubility of finite groups using the sse-embedding property of subgroups. Furthermore, we discuss the sse-embedded subgroups in finite nonabelian simple groups. Some previously known results are generalized and unified.     

Groups with small deviation for non-subnormal subgroups

We introduce the notion of the non-subnormal deviation of a group G. If the deviation is 0 then G satisfies the minimal condition for nonsubnormal subgroups, while if the deviation is at most 1 then G satisfies the so-called weak minimal condition for such subgroups (though the converse does not hold). Here we present some results on groups G that are either soluble or locally nilpotent and that have deviation at most 1. For example, a torsion-free locally nilpotent with deviation at most 1 is nilpotent, while a Baer group with deviation at most 1 has all of its subgroups subnormal.      

The structure of finite groups with trait of non-normal subgroups

In this paper, we investigate the finite groups all of whose non-normal nilpotent subgroups are cyclic. We show that such groups are solvable with cyclic centers. If G is a non-supersolvable group, then G has only one non-cyclic Sylow subgroup which is either isomorphic to Q₈ or is of type (q, q).     

Plancherel identity for semisimple hopf algebras

In this paper we discuss the structure of some product G =AB of nilpotent subgroups A and B. In particular we prove that if G is a minimax soluble group or a finitely generated linear group and if it does not have non-trivial periodic normal subgroups, then G is metanilpotent.     

Finite two-are transitive graphs admitting a ree simple group

ABSTRACT

The article is dedicated to some generalizations of minimax soluble groups satisfying common criterion of nilpotency, such that normality of maximal subgroups, nilpotency of the factor-group by the Frattini subgroup, normality of pronormal subgroups, non-existence of proper abnormal subgroups and so on.     

On a Finite Group Having a Normal Series Whose Factors Have Bicyclic Sylow Subgroups

We consider the structure of a finite group having a normal series whose factors have bicyclic Sylow subgroups. In particular, we investigate groups of odd order and A ₄-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.     

Groups with the weak minimal condition for non-subnormal subgroups

Let G be a group with the property that there are no infinite descending chains of non-subnormal subgroups of G for which all successive indices are infinite. The main results are as follows. If G is locally nilpotent then either G is minimax or G has all subgroups subnormal; if G is a Baer group then all subgroups of G are subnormal. It is also proved that a generalised radical group with this property is soluble-by-finite and either is minimax or has all subgroups subnormal.     

Influence of certain permutable subgroups on finite smooth groups

A subgroup H of a finite group G is said to be permutable in G if it permutes with every subgroup of G. In this paper, we determine the finite groups which have a permutable subgroup of prime order and whose maximal subgroups are totally (generalized) smooth groups.     

On the “Shidov Property” and a Formation Satisfying It

If G is a nonsoluble finite group, the intersection of the maximal subgroups of G which are not nilpotent is the Frattini subgroup of G. This was proved by Shidov (1971 Shidov , L. I. ( 1971 ). On maximal subgroups of finite groups . Sibirsk. Mat. Zh. 12 ( 3 ): 682 – 683 . [Google Scholar]). The authors here present a new formation ? larger than the formation of the nilpotent groups for which the analogous of the theorem of Shidov holds. The theorem makes use of the classification of finite simple groups.     

COMPLETIONS AND CATEGORICAL COMPACTNESS FOR NILPOTENT GROUPS

Abstract

We consider reflection functors in the category of nilpotent groups satisfying certain exactness properties for which the Mal'cev completion functor and the p-cotorsion completion functors are prototypical examples. Each of these functors defines a generalized torsion theory, which in turn defines a closure operator on subgroups. This gives rise to the notion of a categorically compact group with respect to the closure operator which we characterize. This approach provides a unified treatment for the categorically compact groups with respect to the Mal'cev completion and with respect to the p-cotorsion completion, the latter being new. We also consider the p-pro-finite completion, suitably restricted to obtain a reflection functor, and characterize the compact groups so arising.     

Sylow Products and the Solvable Residual

We study the connection between products of Sylow subgroups of a finite group G and the solvable residual of G. Let Π(𝒫) be a product of Sylow subgroups of G such that each prime divisor of |G| is represented exactly once in Π(𝒫). We prove that there exists a unique normal subgroup N of G which is minimal subject to the requirement Π(𝒫) N = G. Furthermore, N is perfect, and the product of all of these subgroups is the solvable residual of G. We also prove that the solvable residual of G is generated by all elements which arise from non-trivial factorizations of 1 _G in such products of Sylow subgroups.     

A Characterization of Nilpotent Injectors in Finite Groups

In this article a class of subgroups of a finite group G, called Q-injectors, is introduced. If G is soluble, the Q-injectors are precisely the injectors of the Fitting sets. A characterization of nilpotent Q-injectors is given as well as a sufficient condition for the solubility of a finite group G, in terms of Q-injectors, which generalizes a well known result.     

Locally nilpotent groups satisfying the weak minimum or maximum condition for subgroups of bounded nilpotency class

We study locally nilpotent groups containing subgroups of classc, c>1, and satisfying the weak maximum condition or the weak minimum condition on c-nilpotent subgroups. It is proved that nilpotent groups of this type are minimax and periodic locally nilpotent groups of this type are Chernikov groups. It is also proved that if a group G is either nilpotent or periodic locally nilpotent and if all of its c-nilpotent subgroups are of finite rank, then G is of finite rank. If G is a non-periodic locally nilpotent group, these results, in general, are not valid.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 384–389, March, 1992.     

On existentially closed and generic nilpotent groups

Letn≧2 be an integer. We prove the following results that are known in casen=2: The upper and the lower central series of an existentially closed nilpotent group of classn coincide. A finitely generic nilpotent group of classn is periodic and the center of a finitely generic torsion-free nilpotent group of classn is isomorphic toQ ⁺, whereas infinitely generic nilpotent groups do not enjoy these properties. We determine the structure of the torsion subgroup of existentially closed nilpotent groups of class 2. Finally we give an algebraic proof that there exist 2^κ non-isomorphic existentially closed nilpotent groups of classn in cardinalityK ≧N ₀. Some results of this paper were contained in [6].     

Groups with a Maximal Irredundant 6-Cover

ABSTRACT

A cover for a group G is a collection of proper subgroups whose union is the whole group G. A cover is irredundant if no proper sub-collection is also a cover, and is called maximal if all its members are maximal subgroups. For an integer n > 2, a cover with n members is called an n-cover. Also, we denote σ (G) = n if G has an n-cover and does not have any m-cover for each integer m < n. In this article, we completely characterize groups with a maximal irredundant 6-cover with core-free intersection. As an application of this result, we characterize the groups G with σ (G) = 6. The intersection of an irredundant n-cover is known to have index bounded by a function of n, though in general the precise bound is not known. We also prove that the exact bound is 36 when n is 6.     

Group algebras of Lie nilpotency index 12 and 13

Let G be a group and let K be a field of characteristic p>0. Lie nilpotent group algebras of strong Lie nilpotency index up to 11 have already been classified. In this paper, our aim is to classify the group algebras KG which are strongly Lie nilpotent of index 12 or 13.     

Torsionsgruppen,deren Untergruppen alle subnormal sind

We prove that a group, which is the extension of a nilpotent torsion group by a soluble group of finite exponent and all of whose subgroups are subnormal, is nilpotent. The problem can be easily reduced to the investigation of extensions of abelian torsion groups by elementary abelian p-groups with all subgroups of these extensions subnormal.     

Pairwise -connected Products of Certain Classes of Finite Groups

Abstract

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.     

ℳ-Supplemented Subgroups and Their Properties

A subgroup H is called ?-supplemented in a finite group G, if there exists a subgroup B of G such that G = HB and H ₁ B is a proper subgroup of G for any maximal subgroup H ₁ of H. In this article, we investigate the influence of ?-supplementation of some primary subgroups in finite groups. Some new results about supersolvable groups and formation are obtained.