Barely Transitive Locally Nilpotent p-Groups

My paper [2], which appeared in the special issue dedicatedto the late Professor Brian Hartley, contained an error in Lemma2.2 as a result of overlooking the case that the exponents k_ion p. 359 may be divisible by arbitrary powers of p, which waspointed out by Professor H. Smith. In order to overcome thissituation all of Section 2, except Lemma 2.1, has been rewritten.However this does not cause any change in the rest of the paper.In particular the original assertions remain correct.     

Factorization of Periodic Locally Solvable Groups by Locally Nilpotent and Nilpotent Subgroups

We establish a series of new results concerning periodic locally solvable and finite solvable groups G = AB with locally nilpotent or nilpotent subgroups A and B.     

Nonperiodic Locally Soluble Groups with Non-Dedekind Locally Nilpotent Norm of Decomposable Subgroups

Ukrainian Mathematical Journal - We study the relationships between the properties of nonperiodic groups and the norms of their decomposable subgroups. The influence of restrictions imposed on the...     

Finite p-Groups All of Whose Proper Quotient Groups are Abelian or Inner-Abelian

In this article, finite p-groups all of whose proper quotient groups are abelian or inner-abelian are classified. As a corollary, finite p-group all of whose proper quotient groups are abelian, and finite p-groups all of whose proper sections are abelian or inner-abelian are also classified.     

On Locally Nilpotent Subgroups of GL 1(D)

In this article we study locally nilpotent subgroups of D*: = GL ₁(D), where D is a division ring. It is proved that every locally nilpotent subnormal subgroup of D* is central. If D is algebraic over its centre then every locally solvable subnormal subgroup of D* is central. Also, in this case, it is shown that every locally nilpotent maximal subgroup of D* can occur as the multiplicative group of some maximal subfield of D.     

On Periodic Locally Solvable Groups Decomposable into the Product of Two Locally Nilpotent Subgroups

We establish new results concerning various properties of a periodic locally solvable group G = A B with locally nilpotent subgroups A and B one of which is hyper-Abelian.     

Commutative Rings whose Proper Ideals are Serial

A result of Nakayama and Skornyakov states that a ring R is an Artinian serial ring if and only if every R-module is serial. This motivated us to study commutative rings for which every proper ideal is serial. In this paper, we determine completely the structure of commutative rings R of which every proper ideal is serial. It is shown that every proper ideal of R is serial, if and only if, either R is a serial ring, or R is a local ring with maximal ideal \({\mathcal {M}}\) such that there exist a uniserial module U and a semisimple module T with \({\mathcal {M}}=U\oplus T\). Moreover, in the latter case, every proper ideal of R is isomorphic to \(U^{\prime }\oplus T^{\prime }\), for some \(U^{\prime }\leq U\) and \(T^{\prime }\leq T\). Furthermore, it is shown that every proper ideal of a commutative Noetherian ring R is serial, if and only if, either R is a finite direct product of discrete valuation domains and local Artinian principal ideal rings, or R is a local ring with maximal ideal \({\mathcal {M}}\) containing a set of elements {w ₁,…,w _n } such that \({\mathcal {M}}=\bigoplus _{i=1}^{n} Rw_{i}\) with at most one non-simple summand. Moreover, another equivalent condition states that: there exists an integer n ≥ 1 such that every proper ideal of R is a direct sum of at most n uniserial R-modules. Finally, we discuss some examples to illustrate our results.     

Hypercentral Groups with all Subgroups Subnormal III

It is shown that a hypercentral group that has all subgroupssubnormal and every non-nilpotent subgroup of bounded defectis nilpotent. As a consequence, a hypercentral group of lengthat most in which every subgroup is subnormal is nilpotent.2000 Mathematics Subject Classification 20E15, 20F14.     

Groups whose Proper Subgroups of Infinite Rank Have a Transitive Normality Relation

A group G is called a T-group if all its subnormal subgroups are normal, and G is a ${\bar{T}}$ -group if every subgroup of G has the property T. It is proved here that if G is a locally soluble group whose proper subgroups of infinite rank have the T-property, then either G is a ${\bar{T}}$ -group or it has finite rank.     

Finite p-Groups with Few Non-major k-Maximal Subgroups

A subgroup of index p ^k of a finite p-group G is called a k-maximal subgroup of G. Denote by d(G) the number of elements in a minimal generator-system of G and by δ _k (G) the number of k-maximal subgroups which do not contain the Frattini subgroup of G. In this paper, the authors classify the finite p-groups with δ_d(G)(G) ≤ p² and δ_d(G)?1(G) = 0, respectively.     

Nilpotent Subgroups of Groups with All Subgroups Subnormal

It is proved that in groups with all subgroups subnormal everynilpotent subgroup is contained in a normal nilpotent subgroup.     

Finite p-Groups with Few Non-major k-Maximal Subgroups

A subgroup of index p~k of a finite p-group G is called a k-maximal subgroup of G.Denote by d(G) the number of elements in a minimal generator-system of G and by δ_k(G) the number of k-maximal subgroups which do not contain the Frattini subgroup of G.In this paper,the authors classify the finite p-groups with δ_(d(G))(G) ≤ p~2 and δ_(d(G)-1)(G) = 0,respectively.     

Addendum: Posets of Subgroups in p-Groups

A finite group is supersoluble if and only if its poset of subgroups satisfies the Jordan–Dedekind chain condition.     

Optimal Subgroups and Applications to Nilpotent Elements

Let G be a reductive group acting on an affine variety X, let x ∈ X be a point whose G-orbit is not closed, and let S be a G-stable closed subvariety of X which meets the closure of the G-orbit of x but does not contain x. In this paper we study G. R. Kempf’s optimal class Ω _G (x; S) of cocharacters of G attached to the point x; in particular, we consider how this optimality transfers to subgroups of G. Suppose K is a G-completely reducible subgroup of G which fixes x, and let H = C _G (K)⁰. Our main result says that the H-orbit of x is also not closed, and the optimal class Ω _H (x; S) for H simply consists of the cocharacters in Ω _G (x; S) which evaluate in H. We apply this result in the case that G acts on its Lie algebra via the adjoint representation to obtain some new information about cocharacters associated with nilpotent elements in good characteristic.     

Finite p-Groups with a Class of Complemented Normal Subgroups

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified.