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Let G be a finite group and H be a Hartley set of G. In this paper, we prove the existence and conjugacy of H -injectors of G and describe the characterization of injectors via radicals. As applications, some known results are directly followed. [ABSTRACT FROM AUTHOR]

In this paper, the structures of dual Lie bialgebras of twisted Schrödinger-Virasoro type are investigated. By studying the maximal good subspaces, we determine the dual Lie coalgebras of the twisted Schrödinger-Virasoro algebras. Then based on this, we construct the dual Lie bialgebra structures of this type. As by-products, four new infinite dimensional Lie algebras are obtained. [ABSTRACT FROM AUTHOR]

Let G be a group and G' be its commutator subgroup. An automorphism α of a group G is called an IA-automorphism if x-1 α(x) ∈ G' for each x ∈ G. The set of all IA-automorphisms of G is denoted by (G). A group G is called semicomplete if and only if (G)=(G), where (G) is the inner automorphism group of G. In this paper we completely characterize semicomplete finite p-groups of class 2; we also classify all semicomplete finite p-groups of order pn (n ≤ 5), where p is an odd prime. This completes our work in 2011. [ABSTRACT FROM AUTHOR]

In this paper, we study the cover-avoid property of 픉-injectors on chief factors of a group G for a Fitting class 픉 in some universe 픘 with partial solubility. [ABSTRACT FROM AUTHOR]

In the present paper we describe a class of Φ-Auslander-Yoneda algebras over K[x]/(xn) in terms of quivers with relations, and prove that they are actually cellular algebras in the sense of Graham and Lehrer. [ABSTRACT FROM AUTHOR]

Let H be an abelian group written additively and k be a positive integer. Let G(H, k) denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sufficient condition for G(H, k1) ≃ G(H, k2). We also discuss the problem when G(H1, k) is isomorphic to G(H2, k) for a given k. Moreover, we give an explicit formula of G(H, k) when H is a p-group and gcd(p, k) = 1. [ABSTRACT FROM AUTHOR]

Let a be an ideal of a local ring (R, m) and M a finitely generated R-inodnle. In this paper we study the Artinianness properties of formal local cohomology modules and we obtain the lower and upper bounds for Artinianness of formal local cohomology modules. Additionally, we determine the set AttRℑdim Ma (M) and we show that the set of all non-isomorphic formal local cohomology modules ℑdim Ma (M) is finite. [ABSTRACT FROM AUTHOR]

Bi-Koszul algebras, including two classes of non-Koszul Artin-Schelter regular algebras of global dimension 4, were a class of graded algebras with non-pure resolutions, introduced in [8]. There is a natural question: can we construct bi-Koszul algebras from algebras with pure resolutions? In this paper, we study this question in terms of normal extensions and Ore extensions. More precisely, we attempt to obtain bi-Koszul algebras from algebras with pure resolutions by these two kinds of extensions. Furthermore, some homological properties of bi-Koszul algebras obtained in such ways are discussed. [ABSTRACT FROM AUTHOR]

For a group G, by ...(G) we denote the number of conjugate classes of the non-cyclic subgroups of G. In this paper, the groups G with ...(G) = 3 are classified. [ABSTRACT FROM AUTHOR]

Let Θ be a collection of formations. A formation 픉 ∈ Θ is called an ℌΘ-critical formation or a minimal non-ℌ Θ-formation if 픉 ⊈ ℌ but the class ℌ of groups contains each proper Θ-subformation of 픉. In this paper we study ℌΘ-critical formations, where Θ is the class of all τ-closed ω-local formations. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of n-isoclinism to the class of all pairs of groups (G,M), where M is a normal subgroup of a group G. The details of that notion are studied and some equivalent conditions are given for two pairs of groups to be n-isoclinic. In addition, the concept of subgroup- and quotient-irreducibility for a pair of groups is defined and we derive how to see whether (G,M) has a pair of subgroups or a pair of factor groups n-isoclinic with (G,M). [ABSTRACT FROM AUTHOR]

A k-nacci sequence in a finite group is a sequence of group elements x0,x1,...,xn,... for which, given an initial (seed) set x0,x1,...,xj-1, each element is defined by \[ x_{n}= \left\{\begin{array}{@{}l@{\quad}l@{}} x_{0}x_{1}\cdots x_{n-1}&\mbox{ for }j \le n 2. [ABSTRACT FROM AUTHOR]

We take in this paper an arbitrary class $\mathcal{K}$ of groups as a base, and define a radical property 풫 for which every group in $\mathcal{K}$ is 풫-semisimple. This is called the upper radical property determined by the class $\mathcal{K}$. At the same time, we define a radical property 풫 for which every group in $\mathcal{K}$ is a 풫-radical group. This is called the first lower radical property determined by the class $\mathcal{K}$. Also, we give another construction leading to the second lower radical property which is proved to be identical with the first one. [ABSTRACT FROM AUTHOR]

In this paper, a new topic about a vast class of subsets of semigroups and binary systems, which contains all ideals, periodic subsets and sub-semigroups, is introduced and studied. In fact, the 'upper periodic subsets' can be considered as a generalization of the conception 'ideals'. We prove a fundamental theorem which states that if A is a (left) upper B-periodic subset of a semigroup S, then under some conditions, it has a unique direct representation $A=\mathfrak{B}\cdot D\,\dot{\cup}\,B^1\cdot E$, where B1=B ∪ {1} and B ⊆ 픅 ≤ S. Especially, we prove a unique direct representation for upper and lower T-periodic subsets, and classify all sub-semigroups of S containing a fixed element T to three classes. This classification gives us more interesting properties for the real semigroups. At last, we characterize upper and lower T-periodic subsets of semigroups and groups. [ABSTRACT FROM AUTHOR]

The paper deals with the following problem: If a finite abelian 2-group is a direct product of its subsets of cardinality 4, does it follow that at least one of the factors is periodic? Two results are presented. In the first one, the structures of the group and the subsets are restricted but the size of the the group is not. In the second one, the group and the factors are general but the order of the group is 26. [ABSTRACT FROM AUTHOR]

Let X be a set with |X| ≥ 3, ${\mathcal T}_X$ the full transformation semigroup on X, and E an equivalence relation on X. Let TE(X) be the set of transformations f in ${\mathcal T}_X$ which preserve E, i.e., (x,y) ∈ E implies (f(x),f(y)) ∈ E. It is known that TE(X) is a subsemigroup of ${\mathcal T}_X$. In this paper, we describe the equivalence relations E so that the semigroup TE(X) is abundant. [ABSTRACT FROM AUTHOR]

The classification of nilpotent Lie algebras of maximal rank was solved by Santharoubane. In the present paper, we prove that the classification of non-degenerate solvable Lie algebras of maximal rank can be obtained from the work of Santharoubane. [ABSTRACT FROM AUTHOR]

Let p be a prime. The class of all p-soluble groups G such that every p-chief factor of G is cyclic and all p-chief factors of G are G-isomorphic is studied in this paper. Some results on T-, PT-, and PST-groups are also obtained. [ABSTRACT FROM AUTHOR]