On Solvable R* Groups of Finite Rank

Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ? S(a) for all 1 ≠ a ∈ G,then ?a ^G ?/?b ^G ? is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.     

On Groups with Given Same-Order Types

On any group G, define g ～ h if g, h ∈ G have the same order. The set of sizes of the equivalence classes with respect to this relation is called the same-order type of G. In this article we prove that a group of the same-order type {1, n} is nilpotent and of the same-order type {1, m, n} is solvable.     

On locally finite -groups satisfying an Engel condition

For a given positive integer and a given prime number , let be the integer satisfying . We show that every locally finite -group, satisfying the -Engel identity, is (nilpotent of -bounded class)-by-(finite exponent) where the best upper bound for the exponent is either or if is odd. When the best upper bound is or . In the second part of the paper we focus our attention on -Engel groups. With the aid of the results of the first part we show that every -Engel -group is soluble and the derived length is bounded by some constant.

     

Engel graph associated with a group

Let G be a non-Engel group and let L(G) be the set of all left Engel elements of G. Associate with G a graph as follows: Take G L(G) as vertices of and join two distinct vertices x and y whenever [x,_ky]≠1 and [y,_kx]≠1 for all positive integers k. We call , the Engel graph of G. In this paper we study the graph theoretical properties of .     

On SS-Quasinormal Subgroups of Finite Groups

A subgroup of a group G is said to be Sylow-quasinormal (S-quasinormal) in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be Supplement-Sylow-quasinormal (SS-quasinormal) in G if there is a supplement B of H to G such that H is permutable with every Sylow subgroup of B. In this article, we investigate the influence of SS-quasinormal of maximal or minimal subgroups of Sylow subgroups of the generalized Fitting subgroup of a finite group.     

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 Remark on the Structure of N-Engel Groups

We observe that there exists a positive integer c(n) such that for every locally nilpotent n-Engel group G we have that G/Z _c(n)(G) is of n-bounded exponent. This strengthens a result of Burns and Medvedev.     

Fibonacci Lengths of Certain Nilpotent 2-Groups

In this paper, we study two classes of 2-generated 2-groups of nilpotency class 2 classified by Kluempen in 2002 and also a class of finite 2-groups of high nilpotency class for their Fibonacci lengths. Their involvement in certain interesting sequences of Tribonacci numbers gives us some explicit formulas for the Fibonacci lengths and this adds to the small class of finite groups for which the Fibonacci length are known.     

On a Commuting Graph on Conjugacy Classes of Groups

We consider the graph Γ(G), associated with the conjugacy classes of a group G. Its vertices are the nontrivial conjugacy classes of G, and we join two different classes C, D, whenever there exist x ∈ G and y ∈ D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Γ(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Γ(G) has at most two components, each of diameter at most 9. If G is any locally finite group, then Γ(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Γ(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes. In particular, we showed that the only nonabelian groups satisfying (i) are the three finite groups of order 6 and 8.     

Second Maximum Sum of Element Orders of Finite Nilpotent Groups

In [1 Amiri , H. , Jafarian Amiri , S. M. , Isaacs , I. M. ( 2009 ). Sums of element orders in finite groups . Communications in Algebra 37 : 2978 – 2980 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], the authors proved that the maximum sum of element orders on finite groups of the same order occurs in cyclic group. In this paper we obtain the structure of groups having maximum sum of element orders on noncyclic nilpotent group of the same order.     

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.     

Centralizers of Involutions in Locally Finite Groups

The present article deals with locally finite groups G having an involution φ such that C _G (φ) is an SF-group. It is shown that G possesses a normal subgroup B which is a central product of finitely many groups isomorphic to PSL(2, K _i ) or SL(2, K _i ) for some infinite locally finite fields K _i of odd characteristic, such that [G, φ]′/B and G/[G, φ] are both SF-groups.     

Minimal Non-FO-Groups

A group G is said to be a “minimal non-FO-group” (an MNFO-group) if all its proper subgroups are FO-groups, but G itself is not. The aim of this article is to study the class of MNFO-groups. The structure of MNFO-groups is completely described, both in nonperfect case and perfect case.     

Infinite Groups with an Anticentral Element

An element of a group is called anticentral if the conjugacy class of that element is equal to the coset of the commutator subgroup containing that element. A group is called Camina group if every element outside the commutator subgroup is anticentral. In this paper, we investigate the structure of locally finite groups with an anticentral element. Moreover, we construct some non-periodic examples of Camina groups, which are not locally solvable.     

The Small Index Property for Free Nilpotent Groups

Let F be a relatively free algebra of infinite rank ?. We say that F has the small index property if any subgroup of Γ = Aut(F) of index at most ? contains the pointwise stabilizer Γ_(U) of a subset U of F of cardinality less than ?. We prove that every infinitely generated free nilpotent/abelian group has the small index property, and discuss a number of applications.     

On the Residual Solvability of Generalized Free Products of Finitely Generated Nilpotent Groups

In this article we study the residual solvability of the generalized free product of finitely generated nilpotent groups. We show that these groups are often residually solvable.     

Constructive Matrix and Orderable Groups

We study into the relationship between constructivizations of an associative commutative ring K with unity and constructivizations of matrix groups GL _n(K) (general), SL _n(K) (special), and UT _n(K) (unitriangular) over K. It is proved that for n 3, a corresponding group is constructible iff so is K. We also look at constructivizations of ordered groups. It turns out that a torsion-free constructible Abelian group is orderly constructible. It is stated that the unitriangular matrix group UT _n(K) over an orderly constructible commutative associative ring K is itself orderly constructible. A similar statement holds also for finitely generated nilpotent groups, and countable free nilpotent groups.     

Groups whose cyclic subgroups have bounded permutability properties

It follows from classical results of Neumann and Macdonald that a group G has finite commuator subgroup if and only if either the normalizers of cyclic subgroups of G have boundedly finite indices or cyclic subgroups of G have bounded indices in their normal closures. In this paper, groups with a similar condition are considered, when normality is replaced by permutability.      

Finitely Generated Nilpotent Groups of Infinite Cyclic Commutator Subgroups

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial Z-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial Z-groups, we finally obtain the structure and invariants of the group G.     

Hypersolvable Groups

Call a group G hypersolvable if it has an ascending series with G/C_G(A) solvable for each factor A of the series. In this article we establish some basic facts about hypersolvable groups. We also prove that if G is a perfect Fitting p-group such that every proper subgroup is contained in a proper normal subgroup, then G has a proper non-hypersolvable subgroup.