A tensor product approach to compute 2-nilpotent multiplier of p-groups

Let [Formula: see text] be a group given by a free presentation [Formula: see text]. The 2-nilpotent multiplier of [Formula: see text] is the abelian group [Formula: see text] which is invariant of [Formula: see text] [R. Baer, Representations of groups as quotient groups, I, II, and III, Trans. Amer. Math. Soc. 58 (1945) 295–419]. An effective approach to compute the 2-nilpotent multiplier of groups has been proposed by Burns and Ellis [On the nilpotent multipliers of a group, Math. Z. 226 (1997) 405–428], which is based on the nonabelian tensor product. We use this method to determine the explicit structure of [Formula: see text], when [Formula: see text] is a finite (generalized) extra special [Formula: see text]-group. Moreover, the descriptions of the triple tensor product [Formula: see text], and the triple exterior product [Formula: see text] are given.

On Periodic Groups Saturated with Finite Frobenius Groups

A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ belongs to a one conjugated to $H$ subgroup of $G$. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

Generalized absolute values, ideals and homomorphisms in mixed lattice groups

A mixed lattice group is a generalization of a lattice ordered group. The theory of mixed lattice semigroups dates back to the 1970s, but the corresponding theory for groups and vector spaces has been relatively unexplored. In this paper we investigate the basic structure of mixed lattice groups, and study how some of the fundamental concepts in Riesz spaces and lattice ordered groups, such as the absolute value and other related ideas, can be extended to mixed lattice groups and mixed lattice vector spaces. We also investigate ideals and study the properties of mixed lattice group homomorphisms and quotient groups. Most of the results in this paper have their analogues in the theory of Riesz spaces.

On Refined Neutrosophic Quotient Groups

This paper is devoted to the study of refined neutrosophic quotient groups. It is shown that the classical isomorphism theorems of groups do not hold for the refined neutrosophic groups. Moreover, the existence of classical morphisms between refined neutrosophic groups G(I1, I2) and neutrosophic groups G(I) is established.

GRUP FAKTOR YANG DIBANGUN DARI SUBGRUP NORMAL FUZZY

A Quotient group is a set which contains coset members and satisfies group definition. These cosets are formed by group and its normal subgroup. A set which contains fuzzy coset members is also called a quotient group. These fuzzy cosets are formed by a group and its fuzzy normal subgroup. The purpose of this research is to explain quotient groups induced by fuzzy normal subgroups and isomorphic between them. This research construct sets which contain fuzzy coset members, define an operation between fuzzy cosets and prove these sets under an operation between fuzzy coset satisfy group definition, and prove theorems relating to qoutient groups and homomorphism. The results of this research are is a qoutient group induced by a fuzzy normal subgroup, where is a fuzzy normal subgroup of a group , is a fuzzy coset, and the binary operation is “” where for every . An epimorphism from a group to a group and a fuzzy normal subgroup of which is constant on cause quotient goup and are isomorphic.

Tópicos de Teoria dos Grupos nos cursos de licenciatura em Matemática

ResumoEste trabalho apresenta parte de uma pesquisa de doutorado em andamento que tem por objetivo investigar quais tópicos da Teoria dos Grupos são imprescindíveis na formação do licenciando em matemática. A pesquisa tem cunho qualitativo, na qual são realizadas entrevistas semiestruturadas com especialistas em Teoria de Grupos, educadores matemáticos envolvidos em educação algébrica e com professores da disciplina em cursos de Licenciatura em Matemática. Neste texto apresentamos os resultados obtidos em duas entrevistas. Os entrevistados concordam que definição de grupo, subgrupos, grupos cíclicos, homomorfismo e isomorfismo de grupos, grupo de permutações, classes laterais, subgrupos normais, grupos abelianos finitos, grupo de transformações no plano e no espaço, grupos quocientes são conteúdos imprescindíveis em um curso de licenciatura. Palavras-Chave: Teoria dos Grupos; Licenciatura em Matemática; Formação de professores.AbstractThis paper presents part of a doctoral research in progress that aims to investigate which topics of group theory are essential in the formation of the graduate in mathematics. The research has a qualitative character, in which semi-structured interviews are conducted with specialists in group theory, mathematical educators involved in algebraic education and with professors of the discipline in undergraduate courses in mathematics. In this text we present the results obtained in two interviews. Interviewees agree that group definition, subgroups, cyclic groups, homomorphism and isomorphism of groups, permutations group, side classes, normal subgroups, finite abelian groups, group of transformations in the plane and space, quotient groups are essential content in a course of degree.Keywords: Group Theory; Degree in Mathematics; Teacher training.