Isomorphism Theorem For Groups Research Articles

On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic Structures

The Jordan–Hölder theorem is proved by using Zassenhaus lemma which is a generalization of the Second Isomorphism Theorem for groups. Goursat’s lemma is a generalization of Zassenhaus lemma, it is an algebraic theorem for characterizing subgroups of the direct product of two groups G 1 × G 2 , and it involves isomorphisms between quotient groups of subgroups of G 1 and G 2 . In this paper, we first extend Goursat’s lemma to R -algebras, i.e., give the version of Goursat’s lemma for algebras, and then generalize Zassenhaus lemma to rings, R -modules, and R -algebras by using the corresponding Goursat’s lemma, i.e., give the versions of Zassenhaus lemma for rings, R -modules, and R -algebras, respectively.

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.

Multiplicative F(m,n)-Hyperrings

In this article, after describing some examples to explain the concept of fuzzy multiplicative (m, n)-hyperrings (F (m,n) -hyperrings), we will offer some related properties. Also, by defining F -hyperideals and regular relations, we will study the quotient of multiplicative F (m,n) -hyperrings. Finally, we will generalize the classical isomorphism theorems of groups to multiplicative F (m,n) -hyperrings.

On wreath product of n-polygroups

In this paper, we shall introduce thin n-subpolygroups of a given n-polygroup and in this regards, the notion of wreath product of n-polygroups will be studied. Also, double cosets of n-polygroups are investigated and the classical isomorphism theorems of groups are generalized to n-polygroups. The main result of the paper is that a finite n-polygroup is singular if and only if it is a wreath product of n-subpolygroups all of which are thin or generated by an involution or by an idempotent element.

Quotient MI-groups

A many identities group (MI-group, for short) is a special algebraic structure in which identity like elements (called pseudoidentities) are specified and collected into a monoidal substructure. In this way, many algebraic structures, such as monoids of fuzzy intervals (numbers) or convex bodies possessing behavior very similar to that of a group structure, may be well described and investigated using a new approach, which seems to be superfluous for the classical structures. The concept of MI-groups was recently introduced by Holčapek and Štěpnička in the paper “MI-algebras: A new framework for arithmetics of (extensional) fuzzy numbers” to demonstrate how a standard structure can be generalized in terms of MI-algebras. This paper is a continuation of the development of MI-group theory and is focused on the construction of quotient MI-groups and a specification of the conditions under which the isomorphism theorems for groups are fulfilled for MI-groups.

KRASNER F (m;n) -HYPERRINGS

$!!!!$ In this paper, the notion of fuzzy $!$ Krasner $!(m, n)$-hyperrings($!F^{(m, n)}!$-hyperrings) by using the notion of$F^m$-hyperoperations and $F^n$-operations is introduced and somerelated properties are investigated. In this regards,relationships between Krasner $F^{(m, n)}$-hyperrings and Krasner$(m, n)$-hyperrings are considered. We shall prove that everyKrasner $F^{(m, n)}$-hyperring is extended by a Krasner $F^{(2,n)}$-hyperring. The concepts of normal $F$-hyperideals andhomomorphisms of Krasner $F^{(m, n)}$-hyperrings are adopted.Also, the quotient of Krasner $F^{(m, n)}$-hyperrings by definingregular relations are studied. Finally, the classical isomorphismtheorems of groups are generalized to Krasner $F^{(m,n)}$-hyperrings provided the $F$-hyperideals considered in themare normal.

Abstract algebra: a comprehensive treatment

An ambitious, comprehensive book covering subject matter typically taught over the course of two or three years, Abstract Algebra offers a self-contained presentation, detailed definitions, and excellent chapter-matched exercises to automatically smooth the trajectory of learning algebra from zero to one. Field-tested through advance use in the ERASMUS educational project in Europe.Including an original treatment of representation of finite groups that avoids the use of semisimple ring theory, Abstract Algebra explains sets, maps, posets, lattices, and other essentials of the algebraic language...Peano's axioms and cardinality...groupoids, semigroups, monoids, groups, and normal subgroups...congruences and isomorphism theorems for groups...cyclic groups and the symmetric group...Sylow's theorems...rings, ideals, homomorphisms, and quotient rings...unique factorization and principal ideal and Euclidean domains...quotient fields and the factoriality of rings of polynomials...modules and vector spaces...Galois theory...linear codes...and ordered rings.

An isomorphism theorem for standard ideals in lattices

Pro@ T is a standard and therefore T is s tandard in H v T . If S ' = ( H A S ) v T is s tandard in H v T then by the Second Isomorphism Theo rem S ' /T is s tandard in ( H v T ) / T which is the first assertion of the theorem. By the Second Isomorphism Theo rem it follows that ( ( H v T ) / T ) / ( S ' / T ) ~ ( H v T ) / S ' . From H v T ~ _ S ' ~ _ T we get H v T = H v S ' and consequently (H v T ) /S '= ( H v S')/S'. By the First I somorphism Theo rem the f ight-hand side is isomorphic to HI(HAS ' ) . Using the fact that T is s tandard we have H A S ' = H A ((HA S) v T)) = ( H A S) v ( H A T) = H A S, i.e. H I ( H A S') = H I ( H A S). Again f rom the First I somorphism Theo rem we conclude H I ( H A S ) ~ ( H v S ) / S , that is, (H v S')/S' ~ (H v S)/S. Thus ( (H v T) /T) / (S ' /T ) ~(H v S)/S, i.e. (*) holds. W e remark that in [1, p. 90 and pp. 252-253] one finds a proof of a common generalization of both isomorphism theorems for groups. This result and the preceding theorem give rise to pose the following