K. P. Shum

Cover-avoidance properties and the structure of finite groups

Abstract We call a subgroup A of a finite group G a CAP -subgroup of G if for any chief factor H/K of G , we have H∩A=K∩A or HA=KA . In this paper, some characterizations for a finite group to be solvable are obtained under the assumption that some of its maximal subgroups or 2-maximal subgroups be CAP -subgroups. We also determine the p -solvability and p -nilpotency of finite groups by considering their CAP -subgroups.

G-covering subgroup systems for the classes of supersoluble and nilpotent groups

LetF be a class of groups andG a group. We call a set Σ of subgroups ofG aG-covering subgroup system for the classF (or directly aF-covering subgroup system ofG) ifG ∈F whenever every subgroup in Σ is inF. In this paper, we provide some nontrivial sets of subgroups of a finite groupG which are simultaneouslyG-covering subgroup systems for the classes of supersoluble and nilpotent groups.

The structure of left C-rpp semigroups

This paper studies the class of left Clifford-rpp semigroups and investigates the structure of their semi-spined products and semilattice decompositions. These semigroups are generalizations of left Clifford semigroups and Clifford-rpp semigroups. We also discuss some special cases such as when a semilattice decomposition becomes a strong semilattice decomposition and a semi-spined product becomes a spined product.

Homomorphisms of implicative semigroups

Implicative semigroups and Brouwerian semigroups were studied by W. C. Nemitz and T. S. Blyth, respectively. In this paper, following the ideas of Nemitz and Blyth, we introduce the notion of negatively partially ordered implicative semigroups and studied the homomorphisms between these semigroups. Some results of Nemitz on implicative semilattices are generalized and amplified to implicative semigroups.

PERFECT rpp SEMIGROUPS

The aim of this paper is to study a class of rpp semigroups, namely the perfect rpp semigroups. We obtain some characterization theorems for such semigroups. In particular, the spined product structure of perfect rpp semigroups is established. As an application of spined product structure, we prove that a perfect rpp semigroup is a strong semilattice of left cancellative planks. By a left cancellative plank, we mean a product of a left cancellative monoid and a rectangular band. Thus, the work of J.B. Fountain on C-rpp semigroups is further developed. *The research of the first author is supported by the Natural Science Foundation and the Education Committee Foundation of Yunnan Province, China and also by the Foundation of Yunnan Province, China. #The research of the second author is partially supported by UGC grant (Hong Kong) #2260126. †The research of the third author is supported by a grant of NSF, China and a grant from the Basic Science Research Council of Yunnan Province, China.

Schur–Zassenhaus Theorem for X-Permutable Subgroups

Let A and B be subgroups of a group G, and ∅ ≠ X ⊆ G. Then A is said to be X-permutable with B if there exists an element x ∈ X such that ABx = BxA. In this paper, a new version of Schur–Zassenhaus theorem is obtained for X-permutable subgroups, and hence, Question 5.1 in [6] is answered.

Markov and Artin Normal Form Theorem for Braid Groups

In this article, we will present the results of Artin–Markov on braid groups by using the Gröbner–Shirshov basis. As a consequence, we can reobtain the normal form of Artin–Markov–Ivanovsky as an easy corollary.

ON THE STRUCTURE OF REGULAR CRYPTO SEMIGROUPS

Superabundant semigroups are generalizations of completely regular semigroups written the class of abundant semigroups. It has been shown by Fountain that an abundant semigroup is superabundant if and only if it is a semilattice of completely J *-simple semigroups. Reilly and Petrich called a semigroup S cryptic if the Greens relation H is a congruence on S. In this paper, we call a superabundant semigroup S a regular crypto semigroup if H * is a congruence on S such that S/H * is a regular band. It will be proved that a superabundant semigroup S is a regular crypto semigroup if and only if S is a refined semilattice of completely J *-simple semigroups. Thus, regular crypto semigroups are generalization of the cryptic semigroups as well as abundant semigroups.

Left Abundant Semigroups

Abstract The aim of this paper is to study some special lpp-semigroups, namely, the left GC-lpp semigroups. After obtaining some properties and characterizations of such semigroups, we establish some structure theorems of this class of semigroups. In addition, we also consider some special cases. As an application, we describe the structure theorems of IC quasi-adequate semigroups whose idempotent band is a regular band.

Isomorphism Theorems for Soft Rings

The concepts of soft rings and idealistic soft rings are introduced. Three basic isomorphism theorems for soft rings are established, and consequently, some properties of soft rings and idealistic soft rings are given.